Thanks both.

I have so far tried:

- Uninstalling and reinstalling the x360 controller drivers
- Starting and restarting the PC with the controller not plugged in
- Changing the stick camera speed in the Control Settings window

No luck with any of them. Note though that changing camera speed does have an effect, it slowed the movement but doesn't stop it outright. This makes sense as my issue is not that the camera moves too much/too fast, but that it moves with zero input from me/XIV considers my 1% misaligned controller stick to be a valid movement.