If you don't mind me asking, what controller are you using exactly? I know your controller supports XInput based on your message, but are you using something like a DualShock 4 with DS4Windows to disguise it as an Xbox controller? I'm not sure why it wouldn't be working otherwise. I apologize, as I typically do not play the game with a controller myself.
I quickly pulled out an Xbox One controller I had lying about and enabled the
XInput hack to test it out. Since it was an Xbox One controller, I went into the
'Mod Settings' option for that particular hack and set the
'Button Names' to
use the Xbox One button names (the 'Mod Settings' option being accessible
here). This step isn't necessary to get the controller working properly with the game, but it's convenient if you're more familiar with the button names the One controller uses instead of the 360 button names.
Using the in-game rebinding menu, I went to 'OPTIONS > CONTROLLER > Character Controls' and set 'Camera Left' and 'Camera Right' to 'C RS Left' and 'C RS Right' respectively:
(You can also do this in the Main Menu where Homer is sleeping on the couch as well. Just keep scrolling until you reach Options. Choose that, then go to 'CONTROLLER', then 'Character Controls' like you would while in-game.)
After doing that, the camera appears to be working fine for me. A right stick left camera input moves the camera to the left and a right input moves the camera to the right.
Note that unfortunately the official PC release of Hit & Run
doesn't let you rebind the pause menu normally. You'd also have to enable the
'Rebindable Menu Gamepad Inputs' hack as well and configure it to your liking using the
Mod Settings button I mentioned earlier.
I hope this information helps!