I've installed DOOM 3 BFG and DOOM 3 from Lutris, then after getting stuck on the copyright screen and getting the screen unresponsive was getting tired of trying, but DOOM 3 (no BFG edition) worked fine, so I gave it another try on Proton issue github and found this bug and found that if I changed the proton version to 4.11-13 the game worked!

For that go to the game on the library > cog wheel > Properties > Compatibility > Check "Force the use of a specific Steam Play compatibility tool > Select "Proton 4.11-13" from the dropdown > Close and Launch

If needed read the github issue there are other fixes that work for some people

Check other versions of Proton if the game gets stuck on copyright screen Check the issue on Proton for this game (BFG Edition) for further fixes