Kind of attack: Attacks (Melee Attacks (Melee Weapon Attacks (Unarmed Strike)))

Kind of weapon: Does not include unarmed strike

So if something says “with a weapon”, it doesn’t apply to unarmed strike. If it says “melee weapon attack” it does apply to unarmed strike (unless it ALSO says “with a weapon”).

Two Weapon Fighting specifically says “with a weapon” so no joy there, even with a feat. The problem isn’t that fists aren’t “light”, it’s that they don’t meet “with a weapon” requirement - even when making a melee weapon attack.

You could use barbarian Battlerager (most DM’s will waive the dwarf only rule) and use medium armor. Great AC, and you can punch as bonus action while raging.

For Heavy Armor fighter, grab Tavern Brawler and use UA unarmed fighting. Then you can punch twice and grapple as bonus action. That’s 2x 1d8 attacks, and a 1d4 if grapple works. That’s pretty darn close to a punch, and unarmed damage on par with monks (possibly better?)