@notthesame I can't believe that worked, you are a godsend. I'd also like to add a few details:
After pressing Start and Y, the LEDs start to rotate from left to right. After that, you hold pair for 2 seconds, the LEDs go off for a second, and then they start to rotate again.
At this point you need to go through the Bluetooth pairing process again (and if your controller is already paired, you should un-pair it before you re-pair it) but you do NOT need to set up the udev rule. All you need to do is reboot and configure the input like normal.
I think this should be added to the documentation.