The game is poorly optimized and full of glitches even on Windows, so it's not Proton's fault that you may experience a lot of problems.

I have installed custom directx config files in the .exe location - they fix slow motion performance and cutscenes lag/audio -, but in order to do so you need to replace the bhd.exe game file to an older version. These files are available in the comments section of this brazilian youtuber's video: https://www.youtube.com/watch?v=EzEOm7PrUDE (I have followed the steps both on Linux and on Windows before and the files are safe. Also, Steam achievements keep working normally.)

From the https://www.pcgamingwiki.com/wiki/Resident_Evil_HD_Remaster article I also recommend installing the Fusion Fix, which enables door skip and has a option in its config file that improves performance.

Without these fixes I was playing the game at 20-30fps, with heavy slow motion. Now I can play it at 55fps and it's very smooth, especially considering my hardware.