WRT WS_EX_NOACTIVATE, there is your patch in Proton currently which does that, it helps this very game with Gamescope (because it won't show a window with WS_EX_NOACTIVATE) but doesn't help on KDE / Gnome. Input focus is only part of the issue, if the window is getting shown on top when made visible (which is probably designed behaviour of this systray window) even without input that still breaks things here.
I know this is not a complete solution, but maybe we may have it to fix at least some cases? Not sure how that can be solved in general without getting rid of this systray popup entirely which would probably be not great as otherwise it is useful.