When you make a melee attack with your pact bow, it counts as an Improvised Weapon that you are proficient with, uses Str or Cha (your choice) as the modifier, and all the pact features still apply, but your DM may rule otherwise.
From the Player's Handbook section on Weapons (chapter 5):
If you use a weapon that has the ammunition property to make a melee attack, you treat the weapon as an improvised weapon.
The bow has the Ammunition property and you are making a melee Attack with it.
According to the rule on Improvised Weapons in the same chapter, the bow deals 1d4 damage if you hit:
If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage.
You are proficient with your pact weapon when using it this way:
See this passage from the PHB entry on the Warlock (Pact of the Blade):
You are proficient with it while you wield it.
You're wielding your bow even though you're making a melee attack, so you're proficient with it. Note that this specific rule may contradict the general rule that characters are not usually proficient with Improvised Weapons, but may be proficient at the DM's discretion:
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
In D&D 5e, the specific rule "you are proficient with it", trumps the general rule "at the DM's option". The introduction to the Player's Handbook has a section titled Specific Beats General:
Remember this: If a specific rule contradicts a general rule, the specific rule wins.
You use Str or Cha as the ability modifier:
The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity.
Making an improvised melee attack with the bow counts as a "melee weapon attack". So your attack and damage modifier is Strength. However, since this character is a Hexblade, the Hex Warrior feature applies:
Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls. This benefit lasts until you finish a long rest. If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon’s type.
So you can use Cha as the modifier, even with the improvised melee attack.
Lifedrinker, Thirsting Blade, and Improved Pact Weapon all apply:
There is no reason that I can see that your eldritch invocations wouldn't apply. E.g. Lifedrinker applies...
When you hit a creature with your pact weapon
That's unambiguous. If your attack hits, you have in fact hit a creature with your pact weapon.
As for Thirsting Blade:
You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.
That also applies. Ditto for Improved Pact Weapon from XGtE. As the adage goes, "the rules do what they say".
But your DM may rule otherwise:
The DM could rule that if you're using your pact weapon in an improvised way, then it's "no longer your pact weapon" somehow.
See also this tweet from Jeremy Crawford (the lead rules designer for Dungeons & Dragons 5th edition):
If you use a weapon in a way that turns it into an improvised weapon—such as smacking someone with a bow—that weapon has none of its regular properties, unless the DM rules otherwise.
If the DM rules this way, then your pact bow is just an improvised weapon that you're not proficient with, uses Strength as the modifier, and none of the pact features apply.
Roll/Hit-wise, it's about a wash. Go human.
You don't really need to keep your bonus action open for anything, so burning it on a weapon butt attack through PAM isn't really a cost.
You're either rolling 2x on 3 attacks or 3x on 2 attacks. The hit rate for Elven Accuracy (EA) means, statistically, you're going to be hitting roughly 25% to 50% more often against a given AC (per any dice) so the net hits will probably end up about the same.
Half Elf Drow gives you Darkness, but you can pick that up as a Warlock anyway. As a damage-focused character, what else are you going to spend your spell slots on?
Human gets 3 attacks at level 3 but Half Elf wouldn't get EA until level 8, so go Human to be more dangerous sooner.
Best Answer
David laid a lot of great foundation here, but I think I found a few ways that we can improve on this a bit.
1) Use a "Hold X" spell. This spell is going to be your best friend. This is because they are paralyzed which means that a) you attack with advantage and b) any attack that hits will CRIT automatically if the attacker is within 5 feet. While gaining advantage and using your Elven-Tri-Vantage and Hexblade's curse is fun, how would hitting a CRIT on an 18, or a 17, or a 16, or a (…) also feel? Because of this, I feel that you can go SUPER-NOVA with this spell.
2) Go Whispers Bard. You can get 2x attacks via Warlock invocations and you basically get a "sneak attack" of psychic damage with Psychic Blades.
3) Not trying to bash at all, but ensure you get your Ability Scores correct here. I get that it says assume that the CHA is 20, but the only way that would happen with David's approach would be to randomly roll your stats and get a 17 or an 18 on the rolls. Otherwise, with David's approach you can only get a +4 CHA with point buy or standard array, so the max would be 361.292 (I think I did that right???...). I have noted that in my builds and have accounted for it appropriately (hopefully)...
With that, I may I present to you
The Hold-CRIT SUPER-NOVA Build
Half-Elf - 13+/x/x/x/x/17 stat priority
Hexblade Warlock 5 (1 ASI)
Paladin 2
Fighter 3
Whispers Bard 10 (2 ASIs)
Round 1: Setup Round
Action: DC 19 Hold Person/Monster
Bonus Action: HBC
Round 2: Kill Round
Action: Attack x2
Action Surge: Attack x2
Bonus Action: Great Weapon Master Attack
You only get this bonus attack if you CRIT (which you do with Hold Person/Monster)
If you succeeded in holding the person, JUST GO HAM.
If you didn't, you have 2 options on your first hit (not at advantage):
Each Attack is as such: (^ => While Supplies Last, you only have 4)
With an additional 1-time per round NOVA of additional damage:
The Numbers
All calculations are assuming that you hit all attacks... didn't have an AC to run this against
50% chance BBEG fails the save with a +8 modifier. Auto CRIT on all hits. That gives us:
50% chance BBEG succeeds save with a +8 modifier. The damage gets a bit more complicated with CRIT-fishing
Now putting All Options together:
But I'm not done yet. While the above build is cleaner, I think I'd prefer to play this variation of it (Does take more setup pre-encounter!).
High Consistency Hold-CRIT SUPER-NOVA Build
NOTE If you don't need Hold Monster, drop a level of Sorcerer and pump bard up to 4 for an additional ASI. This will up the DC of "Hold X", which will push the frequency of you succeeding by about an additional 3% or so.
Half-Elf - 13+/x/x/x/x/17 stat priority
Shadow Sorcerer 6 (1 ASI)
Hexblade Warlock 9 (1 ASI)
Paladin 2
Fighter 3
PRE-BATTLE SETUP
Take 2 Hours & convert Pact Slots into Points and into 4th Lvl Sorc Slots. Each hour will net you 2 4th level slots. This will get you enough slots to smite each time you need to @ max level with divine smite
Round 1: Setup Round
Bonus Action: Place Hound of Ill Omen by target
Action: DC 19 Hold Person/Monster @ Disadvantage (b/c of Hound of Ill Omen)
Round 2: Kill Round
Action: Attack x2
Action Surge: Attack x2
Bonus Action: Great Weapon Master Attack/HBC
You only get this bonus attack if you CRIT (which you do with Hold Person/Monster)
If you succeeded in holding the person, JUST GO HAM.
If you didn't, you have 2 options on your first hit (not at advantage):
Each Attack is as such: (^ => While Supplies Last, you only have 4)
With an additional 1-time per round NOVA of an additional
EXTRA BONUS ROUND: Doggo’s Attack Round
Depending on where the hound is in initiative, it may get 2 of these rounds, but probably only 1
Action: Bite
The Numbers of it all
All calculations are assuming that you hit all attacks... didn't have an AC to run this against
75% Chance BBEG fails this save with a +8 Modifier. Auto CRIT on all hits. That gives us:
25% Chance BBEG succeeds the "Hold X" save with a +8 Modifier. If this happens Cast HBC to help with CRIT fishing.
Now putting All Options together
Cool Notes:
1) An upwards of an extra 25% chance to succeed on your "Hold X" spell which benefits the rest of the group MAJORLY.
1.a) This can be an even greater difference if you have a Rod of the Pact Keeper, as those increase Warlock Spell Save DCs as well!
2) You gain the advantage of a SorLock here, so you can prep for a few hours and gain additional spell slots for “extended NOVA-ing”
2.a) You could even push through a long rest, forgoing your Sorc slots back and risking a point of exhaustion to do nothing but convert Pact slots to Sorc Slots, up to 16 4th level slots!!! Check if your DM will allow it first though.
2.b) My breakdown "Everyday Exhausted" breakdown here would be:
3) You can still HOLD-NOVA without a prep round
Conclusion
I have way too much time on my hands... Hope this gives y'all some more ideas to really push this to yet another level!