Skyscraper
Adventurer
I'm not sure what you mean - battles include roleplaying since you're making decisions your character might also make. Inspiration is just a reward for making decisions consistent with established characterization (specifically, personality traits, ideals, bonds, and flaws from backgrounds). For example, if the soldier fighter has a flaw of "My hatred for my enemies is blind and unreasoning," then rushing headlong into a nasty fight without preparation means the player is roleplaying consistent with established characterization - and that's worthy of Inspiration in my view. If that's a thing one wants in the game, then creating an incentive to do it makes sense. If you don't want the DM to sit in judgment of it (or, if you're like me, and don't want to keep track of 12 to 16 background traits), however, then you can leave it to players acting in good faith to award it to themselves.
I'm all for flavorful RP. What I don't like is that there is an incentive of the DM, namely one of the players around the table; and then the PC (a character) receives a reward in the form of a bonus for another action, simply because the RP by another player was consistent with the PC's background. That's metagaming. You can like that it works this way, but to me it's metagaming.
I also don't like that the player will be conditioned to play along predetermined lines of RP because that's what will get him the candy. I prefer to let the players create whatever they wish.