This right here. I've spent a few hours troubleshooting why I can't play Hell Let Loose, which also uses EAC, even though it should support Linux. Turned out, that you need to specifically search for (in your Library) and install "Proton EasyAntiCheat Runtime", which is a separate game that for some reason didn't get installed when you install the game.
I suppose it's going to be the same with Battlebit, because I'm sure I played it on Linux and had 0 issues.