Been playing Borderlands 3 since it's on Steam now and while not officially supported it does work in proton.
First issue is a Linux one, it has micro stutters, it's not bad and after a few minutes it's almost unnoticeable, this will get ironed out soon most likely.
The second issue is actually due to Windows and 2k Games, it seems they used a video codec (Media Foundation) that is only included with some versions of Windows, why is a video codec in one version of Windows and not another? This was an issue for Pc Build Simulator and Resident Evil, so it's not unheard of. Unfortunately this problem if you encounter it is a show stopper as you can't progress in the game.
It's an easy fix, though the documentation of it is terrible, so here it is step by step what worked for me.
This should work on pretty much any distro.
Obviously you need Steam and the game, my understanding is that you will also need Wine to assist in this install (any version) but it can be removed later.
Download
this and
this (github) as a zip to your downloads folder, then extract both to their own sub folders.
In terminal go to the first directory then run:
WINEPREFIX="/home/{username}/.steam/steam/steamapps/compatdata/{folder number}/pfx" ./mf-install.sh
(use the path above to find out which is the correct folder you need, and note that there may be an extra steam in there, just follow as best you can, you will get there)
Then go to the second folder and run:
WINEPREFIX="/home/{username}/.steam/steam/steamapps/compatdata/{folder number}/pfx" ./install-mf-64.sh
Copy mfplat.dll from cab installer folder into the directory where Borderlands3.exe resides:
/home/{username}/.steam/steam/steamapps/common/Borderlands 3/OakGame/Binaries/Win64/
Now give it a go, mine is using Proton 5.
Of you see Claptrap dance across the screen during intro, it worked.