The question is: Who from among our companions (or Tavs) should take on that role, assuming we want at least one character capable of using this broken feat? Karlach as a Barbarian (Berserker + Thief for the extra bonus action with Enraged Throw), Lae’zel as a Fighter (Eldritch Knight), or a Monk (Sina’zith being the default Monk hireling), given that Monks like to pick Tavern Brawler for increased damage on unarmed attacks? (Of course, if your Tav is a Barbarian, Fighter, or Monk, just insert them into this question.)
It mostly depends on what your party actually is really. Not everyone runs with Karlach and Lae'zel, or uses Hirelings.
Really it comes down to class, Fighter vs Barbarian vs Monk
Fighter is ultimately the best choice, because Improved Extra Attack synergizes with Action Surge and Haste (Outside of Honour Mode) very well.
Barbarian peaks earlier because 5 Berserker + 3 Thief gets you the 2 bonus actions and Extra Attack at level 8 compared to Fighter's Improved Extra Attack at level 11.
Monk has no synergy with thrown items other than wanting to go Tavern Brawler anyway (While Fighter/Barb both can just whack things with melee weapons and ignore Tavern Brawler entirely). But if they're going meta Tavern Brawler Open Hand then throwing isn't even particularly good for them, because they'll simply do more damage with unarmed attacks rather than throwing.
It's worth noting that even though Barb peaks earlier, it's not necessarily that much better than equally leveled Fighter. Since Fighter will still have Action Surge to get off more attacks in a turn which can be all you really need to end a fight. Fighter will also have more Feats due to not multiclassing (As well as being a Fighter, whom gets bonus feats as a class feature) allowing for more utility. Fighter will also have the bonuses of their subclass (EK can get some nice spells such as Shield) while much of Berserker's subclass is just enabling the extra thrown attacks.
and Strength elixirs are finite, too.
They are not. Which is part of why they're so stuipidly broken.
Shops infinitely restock consumables and crafting supplies upon Long Resting, allowing for infinite amounts of Elixirs.
Since enemies like to go for characters with the lowest AC, having Karlach without armour be the tank may make more sense than Lae’zel, who is likely wearing the Adamantine Splint Armour or something similar. Karlach can also take the highest number of hits, since Barbarians are the only class with d12s as their hit-point dice.
Entirely debatable. Enemies will prioritize targets that are threatening them over simply lowest AC much of the time (Due to the simple interaction of Opportunity Attacks)
Lae'zel isn't necessarily wearing Adamantine Splint Armour (Most people actually favour the uncapped medium armour you get in early Act 2 shop and pump Dex for high AC for their "Tank" characters)
Karlach's AC isn't necessarily lower than heavy armour Lae'zel (Especially if you're using Strength Elixirs so you can juice DEX and CON for double AC bonuses from an unarmoured Barbarian Karlach)
Karlach isn't necessarily the one who can take the highest number of hits. Heavy Armour and Heavy Armour Mastery can negate a lot of damage from hits (Even more so when you factor in chugging Health Potions which provide more EHP if you're actively mitigating damage than if you simply have a larger health pool). Throw in stuff like Shield spells and you can mitigate actually taking hits as a Fighter.
In the end, it's largely about how you design your party. Whom you take, what classes you give, what equipment you choose to use.
Overall, I find that Fighter Thrower feels the most practical. It provides a more overall decent character with extra utilities from Feats and Spells as well as early Action Surge can literally wipe out some early fights. The amount of time's I've accidentally killed bosses like Ragzlin, Flind, Anders even Ethel in a single turn of a single Fighter because of Action Surge and obscene throw damage is pretty silly.
But maybe you want your Fighter to do something else (Like Champ multi-crit build. Or Battlemaster) and instead you have no plans for your Barbarian. (Whether or not these characters are Lae'zel and Karlach respectively are another matter entirely. I've often swapped them to alternate "Lore Friendly" class/builds. Personally I quite like 4 Elements Monk Lae'zel a la Orpheus' Honor Guard and also quite enjoy Pala/Fiendlock Karlach playing off her ideals combined with the Zaria enslavement. But I've also done other things like Fighter/Wizard Abjuration Tank Lae'zel, Bard Karlach, Fighter Karlach, Cleric Lae'zel etc. There's a lot of leeway when it comes to creative headcanon to justify alternate classes for Origin characters)