for anyone looking at this in the future, this itch.io release actually does run on the steam deck (even in gaming mode) but requires you to add it as a non steam game and boot it once in desktop mode
Just try it on my deck. Work in desktop mode. Run in game mode, but the controller is not recognized. Is there controller support correctly in the game? Or should I use keyboard emulation in steam?
Really late for an answer. I was already using flatseal for some flatpack permission. Use it to add the folder in the command line, and the controller seem to work correctly. Thanks!