Okay, so, let's start from your wrong premises
Max Damage does not matter - average does
And you are using the wrong weapon for GWF based on that - Greatsword is better than Greataxe. Greatsword is already superior to Greataxe, but it becomes even better with GWF, when the probability of rolling 1s and 2s is higher.
For reference, the average damage from 2d6 with GWF is \$\approx 8.33\$ (you reroll 1s and 2s). The average damage of 1d8 is \$= 4.5\$.
So, for TWF, you deal damage equal1 to $$d_T = (a + 1) \cdot (4.5 + \textrm{Str})$$
where \$a\$ is the number of attacks you have and \$\textrm{Str}\$ is the STR Modifier. For GWF, you deal $$d_G = a \cdot (8.33 + \textrm{Str})$$
We can check when things start going badly for TWF, e.g., for a +5 STR modifier (which, by the way, is not exactly a common roll for a 5th level character, but let's bear with your premise)
$$13.33a > 9.5a + 9.5$$
$$3.83a > 9.5$$
$$a > 2.48$$
So, at 5th level (\$a = 2\$), you are still doing more damage with TWF than GWF. After 11th level, with one more attack, that's no longer true. It gets worse and worse.
Action-Economy
Another thing is: for that extra attack you are using your bonus action. Also note that features like Action Surge will benefit GWF more, since it essentially means \$\tilde{a} = 2a\$, not allowing you to get an extra bonus action. So, for example, in your Barbarian Multiclass, you would be unable to Rage and make the extra attack in the same turn. As the levels go by, you also get more and more uses for your Bonus Action that will not be used because you are using it for an extra attack.
Opportunity Attack
If you are able to make an OA, GWF is superior again, since it's equivalent to increasing your \$a\$ by 1. There isn't much to say here other than that.
Better damage dealing feats
Great Weapon Master exists. And it is insanely good. And I'm not even talking about the extra attack you get on critical or killing hits, I'm talking about the second bullet.
Before you make a melee attack with a heavy weapon that you are proficient with, you can choose to take a -5 penalty to the attack roll. If the attack hits, you add +10 to the attack's damage.
This increases your DPR considerably. On the other hand, TWFs have... Dual Wielder.. which is not as amazing2, sorry.
Stacking magic weapons
I'm not sure what you meant by Stacking magic weapons, but if you have two +3 weapons, they don't give you a +6 bonus.
You have a bonus to attack and damage rolls made with this magic weapon. The bonus is determined by the weapon's rarity.
This means it's actually harder to get magic items as a TWF, because if you want to apply the +3 in every attack, you need TWO +3 magic weapons, while GWF only needs one.
1 Note: the actual damage would be \$p \cdot d_T\$ and \$p \cdot d_G\$, where \$p\$ is the probability of actually hitting. I'm assuming both have the same attack hit modifier (i.e., +5), so they have the same probability of hitting, thus the probability is irrelevant here3. It changes considerably the comparison of DW vs GWM though, thus the math is left aside for that one. If you want the math, I suggest asking a new question about it.
2 Okay, I got a little confused here - I went by OP's words on this one, but then I double checked and Dual Wielder does not give you a reaction or +1 hit/+1 damage. Either way, GWM is usually better than a +2 ASI, which would give you exactly the +1 hit/+1 damage.
After the edit, OP mentions Blade Mastery, but that can also be used with Greatsword, so all it does is increasing \$p\$ for both the styles (see below). As V2Blast mentioned in a comment, Blade Mastery doesn't give you +1 damage either, but let that aside.
3 As noted by nitsua the cool sheep, it's not completely irrelevant because critical hits exist. I'll make the actual math here and let the body as it is. Let \$p\$ be the probability of a normal hit, given by
$$p = (21 - \textrm{AC} + B) \cdot 0.05$$
where AC is the AC and \$B\$ is the modifier to hit, and \$p\$ is capped at 0 and 0.95 (i.e. you always miss rolling 1 and the probability can't be lower than 0). Let \$d_{Ta}\$ be the average damage per attack that hit for TWF, \$d_{Tc}\$ be the average damage per critical hit for TWF, \$d_{Ga}\$ be the average damage per attack that hit for GWF and \$d_{Gc}\$ the average damage per critical hit for GWF. The exact average damage of TWF is given by
$$d_T = (a + 1) \cdot (p \cdot d_{Ta} + 0.05 d_{Tc})$$
and the average damage of GWF is given by
$$d_G = a \cdot (p \cdot d_{Ga} + 0.05 d_{Gc})$$
Choosing your values of STR modifier, we have \$d_{Ta} = 9.5\$, \$d{Tc} = 13\$, \$d_{Ga} = 13.33\$, \$d_{Gc} = 21.66\$. Assuming 14 AC (quite usual), we have \$p = 0.6\$, giving us
$$d_T = (a + 1) \cdot 6.35$$
and
$$d_G = a \cdot 9.081$$
Finding \$a\$ again, we have
$$9.081a > 6.35a + 6.35$$
$$2.731a > 6.35$$
$$a > 2.325$$
So, taking into account critical hits actually makes GWF become even better - because the damage on the dice rolls is larger for GWF.
Best Answer
I have some experience playing an Artificer. I play an Alchemist, not a Battle Smith, but there's a lot of overlap, and a lot of what I've learned applies equally well to a Battle Smith. (Skip to the summary in the last section if you don't want to read the whole thing.)
The Dual Wielder feat is dead weight and should be avoided
If you are still in the process of character creation, I'd strongly advise you to drop the Dual Wielder feat and wield only a single weapon. (If you've already started playing, your DM might be nice and allow you to swap the feat out for another one when you level up.) This feat will have a minimal impact on your damage output, since you already have a bonus action attack available to you through your Iron Defender. You Iron Defender's attack is +4 to hit and deals d8+2 damage. If you forgo this attack and instead attack with a second +1 ylkwa, you attack with +6 to hit and deal d8+1 damage on a hit (since your bonus attack does not add your strength modifier). That's a slight increase in chance to hit and a slight decrease in damage. I haven't done the exact math, but I think the expected damage per round actually works out to slightly less when using two-weapon fighting compared to using your Iron Defender's attack. But regardless of the exact numbers, the difference in damage per round is going to be tiny. In addition, using TWF means that your Iron Defender is barely making any attacks during combat, because your bonus action isn't available to issue commands to it, which isn't very thematically fun.
And what are you giving up to get and use the Dual Wielder feat? First, wielding two weapons would prevent you from wielding a shield, which reduces your potential AC by 1 relative to Dual Wielder. And if you consider a feat as being equivalent in value to an ability score increase, you're also missing out on a +1 to one of your ability modifiers, e.g. intelligence for increasing your attack bonus or dexterity to further increase your AC. I know the variant human feat isn't directly replacing an ASI, but the overall point still stands, since you could also get different starting scores by choosing a different race without the feat. Alternatively, there are any number of other potentially useful feats you might choose instead if you're really keen on playing a variant human.
So the Dual Wielder costs you a lot, because you forgo another feat or ASI and you can't use a shield, and for all of this cost it provides essentially zero benefit and may even slightly decrease your damage output. In addition, it makes your companion much less involved in combat. Hence, I recommend dropping Dual Wielder in favor of something else.
Also note that if you drop Dual Wielder, you have no need for the 2nd ylkwa, which means you also have no need for the 2nd weapon infusion. This allows you to choose a differnent infusion to gain some additional flexibility.
Your "basic attack" sequence is to attack with 1 weapon and your Iron Defender
So, with that out of the way, your "basic attack" sequence becomes clear: use your action to attack with your ylkwa either as a melee attack or by throwing it, and then use your bonus action to command your Iron Defender to attack. There's no dependency between these attacks, so you can make these two attacks in any order. Beyond just choosing targets for attacks, make sure to position your Iron Defender to best use its Defensive Pounce reaction, and try to use positioning to force enemies to attack the Defender rather than you or your party, since you can heal it outside of battle for free using Mending and even revive it for the low cost of a single spell slot.
You probably shouldn't cast Arcane Weapon or Searing Smite during combat
Now, we need to answer the question of when should you should use your bonus action to cast a buff spell like Arcane Weapon or Searing Smite on the first turn of combat instead of attacking with your Defender. By skipping a Defender attack, you lose out on potentially 1d8+2 damage, an average of 6.5. Arcane Weapon and Searing Smite both have the potential to deal an additional d6 damage per round, an average of 3.5. So, at a very rough calculation, you need to get 2 rounds worth of damage out of that spell for it to increase your damage output. However, if you only get 2 rounds worth of damage from the spell, you are barely better than breaking even on damage output, and that damage output is somewhat delayed due to skipping the attack on turn 1. So in that case you've used a spell slot in order to deal almost the same amount of damage over 2 rounds that you would have if you didn't cast the spell. So really, combat must last a minimum of 3 rounds in order for you to get any significant value by casting one of your suggested spells on turn 1. And even over 3 rounds the additional value is pretty marginal. In my experience, most combats don't last longer than 2 or 3 rounds, or at least the outcome is a foregone conclusion by the end of the 3rd round. What this means in practice is that you are actually very unlikely to get any value at all by casting these spells during a typical combat. Most of the time, they will be a waste of a spell slot for no real effect.
Casting either spell before combat is a good idea
On the other hand, if you know that a battle is about to start but hasn't started yet, you can cast either of these spells before the first round of combat and have them already active. In this case, you don't miss out on your first turn bonus action attack, so either spell is a decent option for getting in additional damage starting on turn 1. Arcane Weapon in particular has a duration of 1 hour, so you don't even need to wait until right before the battle to cast it, and you can potentially maintain concentration on it after one battle ends long enough to start the next battle with the spell still active, giving you even more value from the same spell slot.
Note that if you want to get maximum value out of these spells, you will need to maintain concentration on them, which means you want a high constitution modifier. If you decide to drop the Dual Wielder feat, you might consider switching to another race that provides a bonus to constitution to improve your ability to maintain concentration when taking damage, or you could look at the War Caster feat, which gives you advantage on saving throws to maintain concentration.
Also consider Faerie Fire on turn 1
Note that there are several other 1st level Artificer spells with the potential for even greater value in combat. Probably the most relevant is Faerie Fire, which can hit several enemy creatures and give you, your Iron Defender, and all your allies advantage on every attack against the affected creatures. In the right situation, that's potentially a lot more value from a 1st level spell slot than the damage output of either Arcane Weapon or Searing Smite. (The damage increase comes from the increased probability of hitting with advantage, as well as the nearly doubled chance of landing critical hits.)
Putting it all together
So, to summarize: