[GerryC]

The 'reverse' wire should be the center pin on the 7-pin connector. It should have 12V on it when the car in reverse. The ground is the return path.

Since your lamp is an LED, you may be running afoul of BMW's fault monitoring. Like other lamps, it monitors for impedance and current draw on the circuit. The problem with trailers is that BMW doesn't know whether it has incandescent or LED bulbs. If it detects a "fault" it will turn off the circuit to prevent any damage.

One way to check is to do an OBD scan to see if it throws any fault codes. On my F15, it would show an "open circuit" do to the high impedance and minimal current. Some trailers with LEDs will work because they have multiple lamps in parallel that lowers the impedance and raises the current into the proper range.

The solution is to add load resistors to the trailer-side harness. There are solutions and quite a discussion on etrailer.com about it. It is apparently an issue with Audi and Mercedes vehicles as well.

Gerry