Two-Weapon Fighting is best for DPR

TL;DR: The Heavy weapon causes the Halfling to miss too often, losing a lot of DPR and the damage from the second attack of Two-Weapon Fighting is greater than the damage gained from the higher damage die. Eldritch Blast is a very poor option before level 5 if you don't have the Agonizing Blast invocation.

Heavy vs Versatile

First let's compare Heavy weapons with Versatile weapons wielded in 2 hands. A Heavy weapon's 1d12, on average, deals 1 more damage on a normal hit and 2 more damage on a critical hit when compared to a Versatile weapon's 1d10. So, if we multiply these damages by hit chance and crit chance, we can arrive at how much damage a Heavy weapon deals over a Versatile weapon (for a Medium creature).

Regular Hit Formula: \$HitChance=\frac{(20-(Target AC - Attack Bonus))}{20}\$

\begin{array}{c|c} \text{Target AC}&\text{Damage Difference} \\ \hline 10&0.85 \\ 11&0.8 \\ 12&0.75 \\ 13&0.7 \\ 14&0.65 \\ 15&0.6 \\ 16&0.55 \\ 17&0.5 \\ 18&0.45 \\ 19&0.4 \\ 20&0.35 \\ 21&0.3 \\ 22&0.25 \\ 23&0.2 \\ 24&0.15 \\ 25+&0.1 \\ \end{array}

Formula: \$Regular Damage+Critical Damage=1*HitChance+2*0.05 \$

However, Halfling are small, so they have disadvantage on checks with Heavy weapons. This means there is a 95% chance that a critical hit is not one and a chance that a hit becomes a miss. This means the entire damage of the attack (or extra die) is lost, and that damage loss is shown in the following chart:

\begin{array}{c|c} \text{Target AC}&\text{Loss from Disadv} \\ \hline 10&1.83 \\ 11&2.09 \\ 12&2.30 \\ 13&2.47 \\ 14&2.59 \\ 15&2.66 \\ 16&2.68 \\ 17&2.66 \\ 18&2.59 \\ 19&2.47 \\ 20&2.30 \\ 21&2.09 \\ 22&1.83 \\ 23&1.52 \\ 24&1.16 \\ 25+&0.76 \\ \end{array}

Formula: \$Regular Loss+Critical Loss=9.5*HitChance*(1-HitChance)+6.5*0.05*0.95 \$

Since at every relevant AC, the loss is greater than the damage gained from the higher damage die, all other things equal the Versatile weapon is better than the Heavy weapon for the Halfling.

Versatile vs Two-Weapon Fighting

Now that we know Versatile is better than Heavy at all Armor Classes, let's compare Versatile and Two-Weapon Fighting. First consider only the first attack. A Versatile weapon's 1d10, on average, will deal 2 more damage on a normal hit and 4 more damage on a critical hit when compared to a Light weapon's 1d6. We follow the same procedure as with Heavy vs Versatile and see the damage based on target Armor Class.

\begin{array}{c|c} \text{Target AC}&\text{Damage Difference} \\ \hline 10&1.7 \\ 11&1.6 \\ 12&1.5 \\ 13&1.4 \\ 14&1.3 \\ 15&1.2 \\ 16&1.1 \\ 17&1.0 \\ 18&0.9 \\ 19&0.8 \\ 20&0.7 \\ 21&0.6 \\ 22&0.5 \\ 23&0.4 \\ 24&0.3 \\ 25+&0.2 \\ \end{array}

Formula: \$Regular Damage+Critical Damage=2*HitChance+4*0.05 \$

However, When using a Light weapon, the warlock can use Two-Weapon Fighting to get another attack. This other attack doesn't add the ability modifier, but still increases damage by an average of 3.5 on a hit and 7 on a crit. Here is the damage from second attack modified by hit and crit chance:

\begin{array}{c|c} \text{Target AC}&\text{Second Attack} \\ \hline 10&2.98 \\ 11&2.8 \\ 12&2.63 \\ 13&2.45 \\ 14&2.28 \\ 15&2.15 \\ 16&1.93 \\ 17&1.75 \\ 18&1.58 \\ 19&1.4 \\ 20&1.23 \\ 21&1.05 \\ 22&0.88 \\ 23&0.7 \\ 24&0.53 \\ 25+&0.35 \\ \end{array}

Formula: \$Damage=3.5*HitChance+7*0.05 \$

Since, at every relevant AC, the damage from the second attack is greater than the damage gained from the higher damage die, all other things equal Two-Weapon Fighting is better than the Versatile weapon.

Eldritch Blast?

Eldritch Blast is even worse than a Versatile weapon at level 4. Since the warlock does not add Charisma modifier to damage from eldritch blast, the spell loses 3 damage on a regular and critical hit when compared to the Versatile weapon. Here is the Damage Difference between 1 attack of a Versatile weapon and an eldritch blast beam:

\begin{array}{c|c} \text{Target AC}&\text{Damage Difference} \\ \hline 10&2.4 \\ 11&2.25 \\ 12&2.1 \\ 13&1.95 \\ 14&1.8 \\ 15&1.65 \\ 16&1.5 \\ 17&1.35 \\ 18&1.2 \\ 19&1.05 \\ 20&0.9 \\ 21&0.75 \\ 22&0.6 \\ 23&0.45 \\ 24&0.3 \\ 25+&0.15 \\ \end{array}

Formula: \$Regular Damage+Critical Damage=3*HitChance+0*0.05 \$

Since the Versatile weapon is already established as worse for DPR than Two-Weapon Fighting, eldritch blast also falls short (although it does beat out Heavy weapons at low Armor Classes).

Summary Table

I've been doing comparisons throughout this answer to show how flexible math can be for figuring out differences between a few different options saving some time on doing the entire damage calculation. But here is the entire damage calculation for the sake of completeness:

\begin{array}{c|c|c|c} \text{Target AC}&\text{Heavy}&\text{Versatile}&\text{Two-Weapon}&\text{Eldritch Blast} \\ \hline \text{10}&6.41&7.08&8.35&4.68 \\ \text{11}&5.67&6.65&7.85&4.4 \\ \text{12}&4.98&6.23&7.35&4.13 \\ \text{13}&4.34&5.8&6.85&3.85 \\ \text{14}&3.75&5.38&6.35&3.58 \\ \text{15}&3.20&4.95&5.85&3.3 \\ \text{16}&2.7&4.53&5.35&3.03 \\ \text{17}&2.25&4.1&4.85&2.75 \\ \text{18}&1.85&3.68&4.35&2.48 \\ \text{19}&1.49&3.25&3.85&2.2 \\ \text{20}&1.18&2.83&3.35&1.93 \\ \text{21}&0.92&2.4&2.85&1.65 \\ \text{22}&0.71&1.98&2.35&1.38 \\ \text{23}&0.54&1.55&1.85&1.1 \\ \text{24}&0.42&1.13&1.35&0.83 \\ \text{25+}&0.35&0.7&0.85&0.55 \\ \end{array}

Formula: \$Regular Damage+Critical Damage+Second Hit+Second Damage=\frac{Damage Die+1+2*Modifier}{2}*HitChance+2*\frac{DamageDie+1}{2}*0.05+\frac{Damage Die+1}{2}*HitChance+2*\frac{DamageDie+1}{2} \$

note: eldritch blast consists of only a second attack in this formula and the Heavy column has Disadvantage factored in (square HitChance)

Level 5

Level 5 changes things a lot. Eldritch blast gets a second beam, your proficiency bonus increases, and you get access to Thirsting Blade and Eldritch Smite as Eldritch Invocation options if you want to choose them. Since you said you choose Invocations for roleplay purposes, I'm going to assume you did not choose a combat Invocation for this table (if you want to know how to maximize the functionality of one of these invocations, I suggest you ask a new question). Below find the adjusted table for level 5:

\begin{array}{c|c|c|c} \text{Target AC}&\text{Heavy}&\text{Versatile}&\text{Two-Weapon}&\text{Eldritch Blast} \\ \hline \text{10}&7.19&7.5&8.85&9.9 \\ \text{11}&6.41&7.08&8.35&9.35 \\ \text{12}&5.67&6.65&7.85&8.8 \\ \text{13}&4.98&6.23&7.35&8.25 \\ \text{14}&4.34&5.8&6.85&7.7 \\ \text{15}&3.75&5.38&6.35&7.15 \\ \text{16}&3.20&4.95&5.85&6.6 \\ \text{17}&2.7&4.53&5.35&6.05 \\ \text{18}&2.25&4.1&4.855&5.5 \\ \text{19}&1.85&3.68&4.35&4.95 \\ \text{20}&1.49&3.25&3.85&4.4 \\ \text{21}&1.18&2.83&3.35&3.85 \\ \text{22}&0.92&2.4&2.85&3.3 \\ \text{23}&0.71&1.98&2.35&2.75 \\ \text{24}&0.54&1.55&1.85&2.2 \\ \text{25}&0.42&1.13&1.35&1.65 \\ \text{26+}&0.35&0.7&0.85&1.1 \\ \end{array}

Note: the formulae are the same as for the level 4 table, but the eldritch blast column is doubled since there are now two beams

As you can see, without a combat invocation, eldritch blast surpasses the weapon options at level 5 even without Agonizing Blast.

The best choice depends on the situation.

Unfortunately there is no clear winner, so you'll have to weigh the pros and cons. I only included dips (when you pick up a few levels in a class, this is in contrast to builds that split more evenly) and only dual-class multiclassing. It is possible that the best build for your specific campaign and playstyle is a mixture of three or more classes, but this should provide some idea of what benefit you gain from each class.

To start with, we need a baseline to compare to.

Pure Rogue

As a pure rogue, these are the ranges available to you:

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 4&2&6&\text{2d6}\\ 5&2&7&\text{3d6}\\ 7&2&7&\text{4d6}\\ 8&2&8&\text{4d6}\\ 9&2&9&\text{5d6}\\ 11&2&9&\text{6d6}\\ 13&2&10&\text{7d6}\\ 15&2&10&\text{8d6}\\ 17&2&11&\text{9d6}\\ 19&2&11&\text{10d6}\\ \end{array}$$

There is no multiclassing option that can keep up with the Sneak Attack damage here (save perhaps Champion fighter or Hexblade warlock if you can get Advantage often and score critical hits). The reason to multiclass is to increase the Number of Chances to hit or the To-hit Bonus.

Also, staying as a pure Rogue gives you the capstone level 20 ability that guarantees a hit once between short rests.

Now let's look at the options you suggest...

Bladesinger 2

The Bladesinger does not provide any bonuses to the to-hit bonus, the Sneak Attack damage, or the number of chances to hit, but it does increase your ability to maintain concentration on haste. As such, the best time to invest in Bladesinger is after level 13 when you gain access to level 3 spells as an Arcane Trickster. As such, you get this new table:

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 4&2&6&\text{2d6}\\ 5&2&7&\text{3d6}\\ 7&2&7&\text{4d6}\\ 8&2&8&\text{4d6}\\ 9&2&9&\text{5d6}\\ 11&2&9&\text{6d6}\\ 13&2&10&\text{7d6}\\ 15&\text{2 (3 during } haste\text{)}&10&\text{7d6}\\ 17&\text{2 (3 during } haste\text{)}&11&\text{8d6}\\ 19&\text{2 (3 during } haste\text{)}&11&\text{9d6}\\ \end{array}$$

Whether it is worth it to slow down your progression into Rogue to increase your ability to maintain Concentration on haste is something you'd have to decide. Bladesinger and haste also add Intelligence and 2 to your AC respectively.

Bladesinger 5

Alternatively, you could grab haste much earlier from the wizard class. If you collect the wizard levels immediately after becoming an Arcane Trickster, you get this table:

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 5&2&6&\text{2d6}\\ 7&2&7&\text{2d6}\\ 8&\text{2 (3 during } haste\text{)}&7&\text{2d6}\\ 9&\text{2 (3 during } haste\text{)}&9&\text{2d6}\\ 10&\text{2 (3 during } haste\text{)}&9&\text{3d6}\\ 12&\text{2 (3 during } haste\text{)}&9&\text{4d6}\\ 13&\text{2 (3 during } haste\text{)}&10&\text{4d6}\\ 14&\text{2 (3 during } haste\text{)}&10&\text{5d6}\\ 16&\text{2 (3 during } haste\text{)}&10&\text{6d6}\\ 17&\text{2 (3 during } haste\text{)}&11&\text{6d6}\\ 18&\text{2 (3 during } haste\text{)}&11&\text{7d6}\\ 20&\text{2 (3 during } haste\text{)}&11&\text{8d6}\\ \end{array}$$

This option is useful if you expect to be up against the occasional enemy that is very hard to hit (in which case the extra attack from haste is more valuable than the extra damage on only 2 attacks that you might miss), and you don't expect to need the extra damage from Sneak Attack against those enemies that aren't hard to hit.

Bladesinger 6

Putting one more level into Bladesinger gives you Extra Attack, but severely delays the Sneak Attack damage. You get this table:

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 5&2&6&\text{2d6}\\ 7&2&7&\text{2d6}\\ 8&\text{2 (3 during } haste\text{)}&7&\text{2d6}\\ 9&\text{3 (4 during } haste\text{)}&8&\text{2d6}\\ 10&\text{3 (4 during } haste\text{)}&9&\text{2d6}\\ 11&\text{3 (4 during } haste\text{)}&9&\text{3d6}\\ 13&\text{3 (4 during } haste\text{)}&10&\text{4d6}\\ 15&\text{3 (4 during } haste\text{)}&10&\text{5d6}\\ 17&\text{3 (4 during } haste\text{)}&11&\text{6d6}\\ 19&\text{3 (4 during } haste\text{)}&11&\text{7d6}\\ \end{array}$$

This is a good option if the Extra Attack feature is meaningful. Some cases for this include if you are up against a lot of enemies with high armor classes, and if it is beneficial for a magic weapon you collect.

Fighter 2

The reason to dip into Fighter with your criteria would be for Action Surge. Action Surge is very similar in this case to the reason haste is used. Extra chances against hard to hit enemies. However, you can pick it up much sooner than haste, but it doesn't provide its benefit for multiple rounds. Here's the table:

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 5&\text{2 (3 with Action Surge)}&6&\text{2d6}\\ 6&\text{2 (3 with Action Surge)}&7&\text{2d6}\\ 7&\text{2 (3 with Action Surge)}&7&\text{3d6}\\ 9&\text{2 (3 with Action Surge)}&8&\text{4d6}\\ 10&\text{2 (3 with Action Surge)}&9&\text{4d6}\\ 11&\text{2 (3 with Action Surge)}&9&\text{5d6}\\ 13&\text{2 (3 with Action Surge)}&10&\text{6d6}\\ 15&\text{2 (3 with Action Surge)}&10&\text{7d6}\\ 17&\text{2 (3 with Action Surge)}&11&\text{8d6}\\ 19&\text{2 (3 with Action Surge)}&11&\text{9d6}\\ \end{array}$$

This is better than the haste options in Bladesinger in that it requires less investment or levels to reach the extra attack chance, but you get less extra attacks overall. As such, it is best with high AC, but low HP enemies (or if an extra action matters for some other reason). Fighter also grants proficiency in shields if you decide you'd prefer the extra AC to the extra attack with your second weapon. Additionally, if you start with 1 level of fighter, you could have heavy armor proficiency.

Fighter 5

Similar to Bladesinger 6, this is a choice to go for Extra Attack. It requires one less level than the Bladesinger option, and results in the table below.

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 5&\text{2 (3 with Action Surge)}&6&\text{2d6}\\ 7&\text{2 (3 with Action Surge)}&7&\text{2d6}\\ 8&\text{3 (4 with Action Surge)}&7&\text{2d6}\\ 9&\text{3 (4 with Action Surge)}&9&\text{2d6}\\ 10&\text{3 (4 with Action Surge)}&9&\text{3d6}\\ 12&\text{3 (4 with Action Surge)}&9&\text{4d6}\\ 13&\text{3 (4 with Action Surge)}&10&\text{4d6}\\ 14&\text{3 (4 with Action Surge)}&10&\text{5d6}\\ 16&\text{3 (4 with Action Surge)}&10&\text{6d6}\\ 17&\text{3 (4 with Action Surge)}&11&\text{6d6}\\ 18&\text{3 (4 with Action Surge)}&11&\text{7d6}\\ 20&\text{3 (4 with Action Surge)}&11&\text{8d6}\\ \end{array}$$

I didn't include Subclass because there are four choices which all have their own pros and cons.

• Battlemaster for the occasional Precision Attack
• Eldritch Knight to gain access to haste at level 17 instead of level 18
• Champion for the greater chance to score a critical hit with your Sneak Attack
• Samurai to give yourself Advantage if it is needed to trigger Sneak Attack (in case your Dwarf ally isn't next t your target or you have Disadvantage from something)

Now let's get into the ones you didn't list...

Barbarian 2-5

Barbarian is a possible option for a similar reason that Samurai is appealing. Reckless Attack lets you gain advantage at-will (albeit at the cost of granting your foes advantage against you). It follows the same table as the Fighter 2, but without Action Surge. Also, keep in mind, you have to use Strength as your primary ability score to benefit from Reckless Attack.

You can also carry Barbarian to level 5 to gain Extra Attack just like with Fighter, but the subclasses don't provide any boon to your criteria. I'd choose Ancestral Guardian to counteract the Advantage you may grant as a result of Reckless Attack. Barbarian also grants Unarmored Defense letting you add your Constitution to your armor class when wearing no armor. You also gain proficiency in shields from Barbarian in case you want to get +2 AC at the cost of one of your attacks.

War Cleric 2

Undergoing some devotion to the War Domain grants you a Channel Divinity that adds 10 to an attack roll. This can let you hit vital Sneak Attacks throughout the day. Otherwise this build follows the same table as the Fighter 2. The War Domain also offers proficiency in heavy armor.

Vengeance Paladin 3

The Paladin of the Oath of Vengeance has a Channel Divinity that allows you to gain advantage, and as such, has similar benefits to the Barbarian's Reckless Attack or the Samurai's Fighting Spirit. It follows the following table (as would going for just Fighter 3 for the subclasses, except add Action Surge for Fighter):

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 5&2&6&\text{2d6}\\ 7&2&7&\text{2d6}\\ 8&2&7&\text{3d6}\\ 9&2&8&\text{3d6}\\ 10&2&8&\text{4d6}\\ 11&2&9&\text{4d6}\\ 12&2&9&\text{5d6}\\ 13&2&10&\text{5d6}\\ 14&2&10&\text{6d6}\\ 16&2&10&\text{7d6}\\ 17&2&11&\text{7d6}\\ 18&2&11&\text{8d6}\\ 20&2&11&\text{9d6}\\ \end{array}$$

Obviously you could carry Paladin to level 5 to grab Extra Attack and get the same progression as Fighter 5 (but without Action Surge). Paladin offers similar defensive bonuses that Fighter does.

Vengeance Paladin or Fighter 4

In the above case, you could also grab one more level in the paladin or fighter class to get the ability score increase earlier at the cost of a delayed Rogue progression. That results in the following table (add Action Surge for Fighter)

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 5&2&6&\text{2d6}\\ 7&2&7&\text{2d6}\\ 8&2&8&\text{2d6}\\ 9&2&9&\text{3d6}\\ 11&2&9&\text{4d6}\\ 13&2&10&\text{5d6}\\ 15&2&10&\text{6d6}\\ 17&2&11&\text{7d6}\\ 19&2&11&\text{8d6}\\ \end{array}$$

Ranger 3-5

Ranger follows the same progressions as Paladin and Fighter 3-5, but with the following changes depending on subclass:

• Beast Master gets the benefit of another ally that can enable Sneak Attack.
• Gloom Stalker gets an extra attack on its first turn of combat.
• Hunter (Horde Breaker) gets an extra attack if their are multiple enemies adjacent to you and to each other.

Hexblade 1-5

If you expect to be able to get Advantage often, you can delay your Rogue progress by only 1 level to double the number of counts on the d20 that result in critical hits. This is done by grabbing one level of Warlock with the Hexblade patron. It's probably best to take after level 5 so the critical hits hurt that much more. A 1-level dip at level 6 results in this table:

$$ \begin{array}{ccc} \text{Level}&\text{Number of Chances}&\text{To-Hit Bonus}&\text{Damage}\\ 1&2&5&\text{1d6}\\ 3&2&5&\text{2d6}\\ 4&2&6&\text{2d6}\\ 5&2&7&\text{3d6}\\ 8&2&7&\text{4d6}\\ 9&2&9&\text{4d6}\\ 10&2&9&\text{5d6}\\ 12&2&9&\text{6d6}\\ 13&2&10&\text{6d6}\\ 14&2&10&\text{7d6}\\ 16&2&10&\text{8d6}\\ 17&2&11&\text{8d6}\\ 18&2&11&\text{9d6}\\ 20&2&11&\text{10d6}\\ \end{array}$$

You could go a bit further into Hexblade (to level 3) to get the Improved Pact Weapon invocation, which turns one weapon into a +1 magic weapon. This is better if you do not expect to get a magic finesse weapon by around level 6 (which is fairly unlikely using DMG guidelines). It follows the same progression as the other 3-dips, but all but your Two-Weapon Fighting attack gets an additional +1 on the To-Hit Bonus.

After that, you can take Hexblade to level 5 for Thirsting Blade (an extra attack) just like you would Fighter, Paladin, or Ranger. Hexblade also gives you proficiency in shields. See Fighter and Barbarian for details on how this can help with your defensive capacity.

Conclusion

As you can see, every class has something to offer when optimizing for Sneak Attack damage and chances to hit. No one build beats out any other, so you just have to pick and choose what your priorities are. Here are some general notes though:

• Hexblade Warlock and Champion Fighter are the best at Sneak Attack damage when you expect to have advantage.
• Barbarian and Samurai Fighter are the best at getting advantage if your Dwarf friend is elsewhere (Beast Master Ranger provides you an extra melee ally for similar benefit)
• Vengeance Paladin and War Cleric give you the occasional massive boost to your To-Hit Bonus allowing you to hit key attacks for Sneak Attack.
• Gloom Stalker Ranger and Hunter (Horde Breaker) probably grant the most extra attacks, but there are many options that grant you more than the 2 baseline.