EDIT: never mind! I forgot to check the "force the use of a specific Steam Play compatibility tool" in Settings -> Compatibility. With that checked (Proton 8.0.2) it installed and launched just fine (after launching the "repair easy anti-cheat" once)!
It's still working because the game hasn't transitioned from the more lenient EAC to FACEIT yet. About the future of it I've only found a Steam discussion[1] confirming the change will still occur and a youtube pinned comment with a dev quote whose origin I can't find[2]:
"since BattleBit is not competitive as CSGO or R6 etc, the restrictions become overkill for the majority of our player base, because of this, we are still investigating something in between EAC and FACEIT Anti Cheat in terms of requirements"
[1] https://steamcommunity.com/app/671860/discussions/4/67177293...