OK, so here's an update: adding CrossCode as a non-Steam game to my Steam library and then launching it from Steam with Steam Input enabled actually fixed the button mappings. However, the games runs slower like this. Not only that, but I can only aim my shots in a 45-degree cone upwards and downwards. I cannot aim directly right or left, or within 45-degree cones in those directions. This doesn't really seem like intended behavior, either, since the game does normally allow you to fire directly to the sides.