The TCM is responsible for turning the backup lamp relay on and off, so start by making sure the relay is getting a signal from it. Your wiring diagram is on EXL-44 to EXL-46 here:
https://www.nicoclub.com/service-manual ... %2FEXL.pdf
The manual is a little stilted, the connector pinouts are located with the harness instead of the wiring diagram. The backup relay is on PG-115 here:
https://www.nicoclub.com/service-manual ... 2%2FPG.pdf
You can tell if you're on the right relay by the wire colors, it should have Light Green, Red, Brown, Beige wires. LG and BR should have power at all times, BG is the output to the lamps, R leads to the TCM and should have weak power through the coil and then go to ground when you shift to reverse. If it does, the problem is downstream from the relay, if not there's a problem in the TCM wire or the TCM itself.