Based on the stats given, it appears to me that your best option is to not multiclass.
The most obvious option would be a dip into Monk, to get Unarmored Defense's Wis-to-AC bonus, but that would be trading leather's +1 for your Wisdom's +1. The other abilities wouldn't really add anything to what you already have.
Another option is Barbarian's Unarmored Defense. That would add +2 AC from your Con instead of your Leather armor, but simply changing to Studded Leather (also light, and giving full dex bonus) would achieve the same result. The Barbarian's Rage does add a moderately useful ability, but comes at the expense of delaying your progressing in Fighter/Monster Hunter.
Finally, a 2 level dip into ranger would allow you to pick a second fighting style, specifically "Defense" which grants +1 AC while wearing armor. Again, the Ranger's abilities do not synergize particularly with your other abilities, and delaying your main class abilities for 2 more levels is unlikely to be worth +1 AC.
Also note that none of these options combine. Unarmored Defense is specifically called out under multiclassing as an ability you can only get from a single class, and even the greatest Cheeselords still have trouble finding a way to be unarmored and wear armor at the same time.
It seems like you already have the best option figured out: take the Dual Wielder feat for the bonus AC, and indirect damage increase of 2 one-handed weapons.
While you're shopping for a second rapier, upgrade to a suit of studded leather. This and Dual Wielder will bring you up to 17 AC.
After that, just stick with fighter. At 7th level you can start using 2 superiority dice on damage, as well as automatically maxing superiority dice damage against certain types of enemies. At 10th level, your superiority dice get bigger by a step, becoming d10's.
These options will do more for you than you can get from any quick dips into other classes.
For a build like yours, and for most builds in 5e, multiclassing is a great way to increase the breadth of your skills and abilities, but usually does not add a lot to the depth of them.
As Ethan has pointed out in the comments, don't forget to turn to your teammates for assistance and buffs, as well as doing the same for them where you can.
Not effective as a tank
First, his AC in Mage Armor would be 14, as it is not added to Unarmored Defense, but replaces it. You will have to rely on rage to be tanky.
In rage
He would take half damage from some damage types, but could only do weapon attacks. Can't even concentrate on Hex, which is about the only reason why Warlock based warriors are competitive.
Outside of rage
He is just a somewhat less powerful Warlock
Only benefit
Con saves are arguably more important than Wis saves for casters, because of concentration. Str and Cha are about the same.
Better alternatives
Two levels of Paladin adds Heavy Armor, great spells, a fighting style and Divine Strike.
One level of Fighter adds Heavy Armor, Con save and a fighting style.
Armor cost
Do not worry about the cost of armor, soon you will have too much gold.
Best Answer
The core idea of this build seems to be: you want to have a high AC and be a tank, but you only need to spend two levels on paladin to pull that off, and you're going to spend the rest of your levels on something with higher utility. I haven't tried this myself, but I've seen others try it and it seems to work pretty well.
The thing where you heal yourself as a warlock seems like a good trick. It's only a tiny bit more healing than you would have gotten from Lay On Hands as a normal paladin, but having it be a bonus action is useful.
Smiting using your warlock spell slots is also a good trick.
You've written that you want to have a high damage output to draw aggro. I don't think this will work as well as you want, because being a multiclass character will drop your damage output significantly. In particular, a single-class paladin would get Extra Attack at fifth level, and you're not going to get that until much later.
You'll get shadow blade instead, but it's not as good: the damage output is less (2d8+DEX with your build, versus two attacks at 1d8+DEX each for the paladin), you're spending spell slots which could be used for other things (like smites), and every time you get hit, you'll have to make a constitution save or lose your weapon.
(Also, tanks don't hold aggro by having high damage output. Tanks hold aggro by having the Sentinel feat.)
As a side note, it's not surprising that it's hard to optimize both AC and damage simultaneously. If there were a build that gave you high damage and also high AC, everyone would be doing it already.
As Andras notes, your current build requires STR 13 and a high DEX. If you switch to a STR-focused build, then your stat requirements go down and your AC goes up, at the cost of some initiative. I'm not sure what your stat array looks like, but one guess might be that you'd go from 13/15/12/8/10/14 to 15/8/13/10/12/14, and you could put a racial +1 in CON to get bonus hit points.
I don't think Agonizing Blast is a good choice for you. Your eldritch blast spell has a low attack bonus because CHA isn't your primary stat; also, it will have disadvantage any time you use it while in melee combat. Warlock invocations are valuable, and it seems weird to spend one for a minor buff to an ability you'll rarely use.
I'm unsure about Grasp of Hadar. You might be able to use it to pull foes out of the air, or to pull them away from allies they were grappling. Those are good effects.
But the thing is: you're also spending an ASI on this (for War Caster). There are lots of super valuable things you could be doing with an ASI and two warlock invocations. The ASI could give you +1 to attack and damage rolls; the invocations could add non-combat utility to your character with Beast Speech or Mask of Many Faces or Misty Visions, or you could just get the Devil's Sight invocation and use the darkness spell.