I installed the game via Proton. Seems it is still very buggy in wine. I had to
follow the instructions here [1] to get it to load without crashing or
constantly changing the screen between windowed and fullscreen. Once I got it
working the menu was working fine, no lag.

I'm suspecting you are either experiencing a wine bug or you have some bad
settings somewhere rather than a Mesa bug.

[1] https://www.protondb.com/app/39510