[RPG] What are the ways the non-monk character can improve their unarmed damage

dnd-5eoptimizationunarmed-combat

My Goliath Barbarian / Fighter / Rogue hybrid killed somebody accidentally in a rage in their past, and has sworn an oath to never again wield a weapon to prevent accidentally killing somebody again if they fly of the rails.

Problem is, this is still D&D, and punching people for 1+strength mod+2 while raging is hardly going to impress anybody. He'll primarily be a grappler, so it's not that big a deal, but I'd still like to find some ways to improve his damage.

There's a caveat however, I can't take any feats, considering my classes are spread all over the place and I need the few attribute increases I can get.

These are the requirements:

  • Increases can come from magic items. I've found the Insignia of Claws in Hoard of the Dragon Queen, but that's the only one I've found so far.
  • Can't change my race into something else. I know Tabaxi, for example, have improved unarmed damage, but that doesn't help my Goliath.
  • Can't multiclass into monk or any other class. I just don't have the levels left to start multiclassing into a 4th class, nor would I meet the wisdom requirement.
  • No feats. I know tavern brawler will increase my damage to 1d4, but I don't have room for feats in this build because of its multiclass nature.
  • DM will not allow sneak attack on unarmed attacks.
  • The build so far is 4 levels barbarian (Bear totem), 3 levels fighter (Battlemaster), 1 level rogue. The rest of the levels will go into fighter and barbarian, so they offer no leeway for multiclassing.
  • Other people casting buffs on the character is a valid advice.
  • I'll likely be grappling two people at the same time, so using 'not a weapon' improvised attacks is not an option, it's going to be kicks and headbutts.

Best Answer

If you can get your DM to allow use of Unearthed Arcana: Class Feature Variants, you can make use of Martial Versatility (replace Fighting Style with another on levelup) and Unarmed Fighting style, which works well with grappling:

Your unarmed strikes can deal bludgeoning damage equal to 1d6 + your Strength modifier. If you strike with two free hands, the d6 becomes a d8.
When you successfully start a grapple, you can deal 1d4 bludgeoning damage to the grappled creature. Until the grapple ends, you can also deal this damage to the creature whenever you hit it with a melee attack.

So, in the end you would deal d6+d4+str+rage damage to a grappled creature for 5 average damage per hit increase from what you have now.

EDIT: Unarmed Fighting was printed in the Tasha's Cauldron of Everything with one difference: instead of damaging enemy on grapple start and every hit, you instead deal automatic damage to a single creature at the beginning of your turn.