Even what she should mechanically take as a feat. Still leaning towards Tavern Brawler - even though it's dumb, oOoO, I should probably take 20 CHA instead~
It's dumb, oOoO, you should probably take 20 CHA instead.
The grappling parts of Tavern Brawler aren't even that good from a grappling perspective, and are completely useless to a non-grappler. The unarmed damage die upgrade is nice if you plan on punching people a lot, which is the
real reason grapplers might pick the feat up, but if you're not an extremely punchy character it's not going to do much either.
As for proficiency with improvised weapons, you already have proficiency with improvised objects that resemble weapons you're proficient with. I mean, this is technically a "GM's option", but I'm going to go ahead and assume that your particular GM isn't the kind who would say you can't use a lump of wood or table leg as a club (which you are proficient with) because it's not exactly the kind of lumpy wood that a club is designated as being.
One thing is trying to argue that the cool rock you picked up should be treated as a double-bladed scimitar, it's entirely another to say "this whacking stick is a whacking stick".
Basically the only thing you'd directly get out of that is proficiency with using
specifically your lute as a weapon, and as a dabbling musician I can only scream silently at such a decision.
EDIT: If you
really want to drive home the "cheerleader" role, you could potentially take something like Martial Adept and pick up a superiority die. There's no stat bonus attached to the feat (and 20 CHA really is quite nice), but you get access to things like Commanding Strike or Maneuvering Strike, which would let you play fangirl to the Dragopal by giving them more things to do with their reaction (there's also Rally, but it's a piddling amount of temporary hitpoints and temp HP doesn't stack with any other source of temp HP, of which there are many). As far as bar room brawl applicability is concerned, most maneuvers require only a "weapon attack", which includes unarmed strikes and throwing rocks/mugs at people.