It seems the issue from this thread has reared its head again.
Each start of bnet, WOTLK classic client queues an update. The update is very slow, occasionally fails, but ultimately lets the game start (often even if it throws an error). Without closing bnet or even the game, at some point it decides an update is required again. So there’s just no fast way to restart the game at present.
Same as with that post above, older versions of WINE work around it. But the general performance hit many seem to be experiencing after the Ulduar patch is just exacerbated when using the old versions of WINE and Proton. Pre-ulduar I could maintain 200+ FPS but set a cap, now it drops to 10-15 during raid encounters on older WINE or Proton, and just manages to stay above 50 fps or so on updated WINE/Proton. So new WINE or Proton is still struggling a for sure, but it’s a necessity to not be dead weight. Perpetual updates caused by using it which turn launch times usually measured in seconds into minutes isn’t ideal though.