Backup lights are in the trunk lid, right? Whenever a light in the trunk lid fails, the first suspect is fatigue and breaking of the wires that pass through the hinge to the trunk lid. However, I would expect both backup lights to be powered by a single wire running through the hinge area. You will need to detach the trunk lid liner and check. If there are separate wires, pull back the shielding around the wires at the hinge and examine/feel them for evidence of breaking at the bend point.
For some reason, BMW does not expect the trunk to open as often as the doors, so they use standard multistrand wire in the trunk hinge, rather than the long-life flex wire that they use in the door hinges.
It is possible that there is a corrosion problem with the plugs in the trunk lid tail lights like the known problem with the outboard tail light assemblies.