[RPG] Does the Martial Adept superiority dice scale up alongside the Battle Master fighter’s superiority dice if you acquire the subclass after the feat

Yes

"If the attack involves other damage dice" sounds like a pretty solid indicator that "Superiority Dice" would be multiplied, if a maneuver were used that adds the result as damage.

While not applicable to everybody, that sounds like an awesome way to add extra damage to multiply (not that superiority dice are really game-breaking for additional damage) after learning that an attack hit... but I'd allow it, regardless.

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.