None of the lights on the car have anything to do with the backup lights, not one part mixes with the lighting system on the car.
The backup lights are part of the WIPER circuit. Odd as that sounds.
Do yourself a favor and print out the wiring diagrams and trace the circuit, just the circuit needed. I do this anytime I'm troubleshooting which allows me to follow the current through the circuit.
(The blue highlighted wire is the white on the car)