I use a PSone dualshock controller, hooked up with a USB converter, to play this game. One quirk it seems to have is that the default keybindings are a little out of whack with what I think they're supposed to be (e.g. the action buttons are mapped mirrored compared to how they're laid out on the HUD, the camera axes are switched with each other so up and down turns the camera, etc.).

This wouldn't be a real problem, except I apparently have to set them again each time I open the game. Does anyone have an idea why my bindings wouldn't persist past the session I set them in?

Try running the game explicitly as Administrator by right-clicking its shortcut and selecting the "Run as" option.
Don't install the game into Windows' Program Files folder, but into a custom location on C: (e.g. C:\GOG Games\)

I'll have to check for sure when I get home, but as far as I remember it's already set to run as admin (I get the notification everytime I start the game up; it was like that since I first installed it).
Just in case it turns out to not be this, is there any other likely cause you can think of?

Also, just to clarify, the game is in a custom location on a non-system partition, and not in Program FIles.