Thank you for the advice. I did try to change the runtime a couple of times before, but I don't know for sure if GE-Proton was one. I tried using Lutris again like I did before so I could select GE-proton, but this the installer would not even finish. Since I use GOG also, I decided to try the Heroic Launcher flatpak, and that installed Battle.net just fine. I'm now downloading WoW and I'll keep you posted how that goes.
Edit: that worked! No noticeable issues so far.