The issue is that full screen games wouldn't hold onto the mouse? Are these games running through proton (windows games being launched through steam)? If so, I know there's an option in protontricks to tweak this behavior per game. "Automatically capture the mouse in full screen windows" in winecfg.

Alternatively, you can try tweaking your steam launch params to use gamescope. Ex.

gamescope -W 1920 -H 1080 -r 60 -- %command%

Where the params denote the resolution and refresh rate of the window. You may need to install gamescope from your package manager.