I spent way too long on this deceptively simple problem. 
I assumed there would be a clean setting somewhere to just force “gamepad and keyboard only” input, but nope—not quite that easy.
In the end, I added mouse click events to the Widget itself and just manually re-focused the last known button whenever a stray click happened. It’s a bit janky, but definitely better than the menu losing focus and trapping the user in UI limbo.
Hopefully this helps someone else avoid the rabbit hole!
