Tavern Brawler
Accustomed to rough-and-tumble fighting using whatever weapons happen to be on hand, you gain the following benefits:
- [...]
- You are proficient with improvised weapons.
- [...]
Now, this would technically turn a Shield into an improvised weapon, with which you are proficient, that deals 1d4 damage. Does this make the Fighter's Duelist Fighting Style incompatible with a Shield if you get the Tavern Brawler feat?
Duelist (Fighting Style)
Gain +2 bonus to damage rolls when wielding a melee weapon in one hand and no other weapon.
(Emphasis mine.)