I'm currently on openSUSE Tumbleweed with Lutris installed as a flatpak from flathub.
The game runs pretty decently though I had to turn down the graphics a bit compared to what I can run in Windows.
After a while tho the game starts chugging and then basically freezes almost completely. I can still hear sounds in the background but the game looks like it slows down to like one frame per minute and I can't tab out or switch over to a terminal with alt ctrl F* either. Basically my entire computer slows to a halt and I end up having to do a hard pwoer off.
I have it installed through Lutris in the same way you have, running on Arch instead but that should make no difference really.
I don't have those problems at the moment.
Perhaps your setting within Lutris are not right?
Which version of proton are you running?