There is a way to make full screen actually work properly (in this and Hempuli's other MMF games), but it involves running xdotool to force-resize the game "window" to the display resolution. I can put up my script somewhere if that would be helpful.
I also made a Steam Controller (yeah, the OG) layout that uses absolute mouse positioning to map controller buttons to the on-screen Undo/Restart/etc. buttons, alongside the standard mouse controls. It requires the above fullscreen fix though (in fact it was the motivation for me to fix it in the first place), because Steam Input mouse positioning is broken if the game window is upscaled at all. Shockingly enough, Steam allows you to share controller layouts for non-Steam shortcuts and have them show up for others if they name their shortcut the same thing (at least I think that's how it works?). So if you make sure your shortcut is named precisely "Mobile Suit Baba" (no quotes ofc), then once I upload the layout you should be able to find it? (I didn't do the on-screen-button stuff for the Deck controls since I found the touchscreen to be suitable for them, I just used the trackpad on it otherwise. But you can apply the SC mapping to the Deck just the same.)
I do agree that an actual controller input mode would be nice to have, though. It's just something that requires a fair amount of work. But even keyboard shortcuts for the clickable buttons would go a long way, I think.