Simplest way I think is saving the controller layout as a template and then you should be available to see in all games when you go to templates.
I do this for the Steam Controller where I saved a general template to use across games so I don’t need to reconfigure every single part from scratch.
Do you encounter the same issue in desktop mode too?
I tried it on my steam deck and was able to apply my custom template to another game in the normal Steam Deck mode so I wonder if maybe the config is corrupted? Maybe make a simple test config template to check out if it applies to other games is worth trying before making your main config from scratch again.