I have always been able to diagnose issues through posts found on this forum but this one I can't resolve.
I have a 2010 Mini R56, the sunroof drain was clogged and this caused my FRM module to go bad. With help from the folks at ECU Pro I replaced the module and for the most part everything works.
The only thing I am having an issue with now is my rear right reverse light which does not turn on when I put the gear in reverse. The only reason I noticed the rear right reverse light does not work is because I get a "bulb" message icon on my screen every time I turn the car on.
I checked the reverse right light bulb and it works fine.
I put the left light assembly on the right side (works when on the left side) and the reverse light does not work. This confirms its not the bulb or the assembly.
I checked the fuses and they all appear to be good.
I would exclude the reverse switch since the left reverse light works.
I tried to unplug the FRM module and clean all the plugs, I also cleaned al the plugs at the taillight assembly plug but still nothing.

Any suggestions would be very much appreciated.

 

A clogged sunroof drain cause an FRM failure?

My wife's 2014 X3 does the exact same thing with the reverse lights; I even performed the exact same steps as you. I can connect my Foxwell scan tool, bring up the active tests under the footwell module, and turn on the reverse lights with the bi-directional control. The reverse lights work fine, but I intermittently get a bulb-out warning. BMW's burnt out bulb software needs some type of software patch. Richard at Fairfield imports told me once that with every BMW/MINI he sells he goes into the programming and turns off the bulb out warning. The engineer must not have graduated top of his class, so they kept him away from the engine and transmission department and said "Here ... you go make a bulb out warning program."

What I did learn from the X3 is that the FRM controls each of the reverse lights independently (separate circuits).

 

It did in my case. I was away for travel and the car remained outside for a couple of months. When I returned the carpet was all damp. I raised it only to find a puddle that took weeks to dry.
The FRM module pins were all corroded.