Oh, that seems to do the trick. I'm getting up the Blizzard install window now at least. Still installing though, but I'm getting further than usual.
Edit: Until it was done and I tried to login. Keep getting
Battle.net Helper.exe has encountered a serious problem.
Can't enter the password either, not sure if the problems are connected.
Edit 2: Both problems solved by changing Wine version FROM
System (2.10) (Default) TO
Edit 3: Could login once, but now my login Window looks weird (Didn't do that the first time). I can hit login, but the authenticator window which shows up doesn't send me the auth code this time when changing to SMS.
Edit 4: I'm able to play if I chose "Continue offline" and login directly thru the game. However, I disabled my SMS authentication as I could not get it to work. The phone app works though.