TLDR: I did the same procedure as Phattie and it seems Starcraft II works again and can be updated fine for me now, thank you very much for this.
As I had uninstalled it while troubleshooting I had to reinstall it using the lutris installer:
When the battlenet installer opens I go into drive_c/users/<myname>/Application Data/Battle.net/Battle.net.config as described by Phattie’s comment above and set HardwareAcceleration and StreamingEnabled to true.
When the battle.net installer does the 100%->0% thing and stops forever I waited a bit and then killed the battlenet launcher install process (ps aux | grep .exe then kill the installer one) which lets the lutris installer continue.
After this I set the wine version to proton and started it up. The launcher started normally and I could install Starcraft II. I have now launched the game and it works perfectly.
Notes on this method:
The game install window takes some time to open after clicking the button so don’t worry about nothing happening immediately.
The download progress visually stays at 0% until it reaches playable and then stays at that percent. It stopped at 93% for me but after I restarted battle.net it said it was fully installed.
I get some random error messages periodically while battle.net is running but I just close it immediately after starting Starcraft so it doesn’t bother me.
It seems like it sometimes gets stuck at “trying to wake up” when starting the client but turning it off and on again fixed it every time it occured for me.
I personally did not have to replace any files with older versions or do anything else that chazdingle mentioned but that doesn’t mean it works the same for everyone of course.