My Switch Pro Controller works fine on desktop and with Steam, but in-game it goes completely crazy. Constant random button presses make it impossible to do anything, and I obviously can't edit the bindings while the controller is plugged in.

Assuming you are using Windows, install DS4Windows: https://github.com/Ryochan7/DS4Windows/releases/tag/v3.0.18
It tricks Windows into thinking your Swtich controller is an Xbox controller. This is the prefered controller to use if playing on Windows.

You can now use Switch Pro Controller using either DS4Windows or GlosSI. You can read the guides here:
https://mrcrayfish.gitbook.io/controllable-documentation/guides/how-to-connect-a-gamepad