Hey thank you! I can confirm that XInput gamepads should work correctly but I can't guarantee all models will function the same. I tested the game with Xbox controller and my colleague tested with PS4 controller (with the help of a plugin iirc) because DirectInput has some compatibility issues. (I have one of those bluetooth gamepads that are meant to be used with android devices, and it doesn't get recognized by the game)
At the moment there isn't a way to customize buttons except changing Types, but it doesn't affect things like movement and look. I'd like to make a customizable controls for sure, definitely for the full game. Sorry I can't help much but have you tried using any PS controller plugins?