I'm using the latest GE Proton as well, installed Battle.net through the New Game/App dialog. Just found out that Battle.net does start, but it simply never shows a dialogue and therefore there is no way to interact with it. No taskbar item as well of course.

Solving vague issues with Linux systems is already a big part of my day job and last week was pretty rough, so basically the choice boils down to :

• Spend energy I honestly do not have on whipping Faugus into shape.
• Keep using a working Lutris setup, even though it is AI slop.

Neither appeal to me at the moment. 😞

Edit: the irony, just as I was about to give up an update for Faugus appeared, installed it and Battle.net now runs fine. 🎉