WineGE is old and dead. Don't use that. Some other things:
- You sure the game is set to Fullscreen and not "Borderless Window"? There's a big difference.
- Try to ALT+Tab once in and out of the window to see if it snaps back into focus.
- Try Wine 10+ instead. Perhaps a regression was fixed
- Try Proton instead of Wine if using Lutris. Works fine on Deck.