As "F30" states, the E91 tends to develop broken wires at the tailgate hinge plane, inside the black gaiters or covers of the individual wires. EVERYTHING electrical on the tailgate is dependent upon wiring inside those gaiters, including rear Wiper/Washer, Reversing Lamps, inner tail lamps, license lamps, the "Aerial Diversity" that is related to receiving Remote Key signal to unlock & radio reception, and Tailgate Latch which is electrically-operated.
There could be a break in the Gray/Brown 12V+ power supply wire, or in the Brown ground wire, as shown in the following circuit diagram for the License Plate Lights at far-right of schematic:
https://www.newtis.info/tisv2/a/en/e...-light/i7o2pTr
I would suggest checking for BOTH 12V+ voltage at pin #2, and for continuity to CHASSIS ground (NOT tailgate metal) at pin #1 of BOTH light connectors, X709 & X710 per this Installation Location:
https://www.newtis.info/tisv2/a/en/e91-325i-tou/RqhAbp0
Without actual electrical tests with multimeter (and test of bulb by applying 12V+ & ground to the pins) one can only guess. The "Bulb Fail Monitor" tests for current flow in the system so ANY open circuit, including bulb filament, broken/disconnected ground wire, or break in 12V+ wire will cause bulb failure warning.
It can ONLY detect an open circuit somewhere between Pin #49 of FRM Connector X14261 and Chassis/Battery Ground, and CANNOT locate the fault definitively. It SHOULD be thought of/ described as a "Bulb CIRCUIT Monitor."
Please let us know what you find,
George