Denki - you may need to check which version of Unity you are currently using because I have just spotted a blog about an unrelated game that had the "Linux Fullscreen breaks Mouse Input" problem and it was attributed to a bug in Unity that has apparently been addressed.
Here is the link to the blog where the issue is presented with an interim fix for Steam Linux users until Unity was changed to the new version...
https://www.gamingonlinux.com/articles/verdun-is-yet-another-unity-title-with-br...