BookBarbarian
Expert Long Rester
I wouldn't be too concerned about feats vs styles. But I do agree that it conflicts with Tavern Brawler. I don't much like TB, that could be revised too, but I guess we're looking for a minimal construction here.
How about when you have a free hand and succeed on a grapple or shove, you also deal your weapon damage die? (Without any bonuses.)
Yeah, that sounds pretty good.
Or maybe something like adding the weapons damage die to the Athletics check to Grapple? (Edit: that might be too strong so Weapon die roll divided by 2?)
Anyway I like the direction your going.