Yep, you need the OBD to round connector adapter. Are you getting power at the connector behind the light? Also I'm pretty sure the reverse light power feed does not go anywhere near the instrument cluster. The LCM is the unit that controls all the delivery of power to the various lights around the car.
With INPA software you could check for errors specific to those on the LCM & you can also simulate the feed from the switch on the gearbox to activate the reverse lights.
My bet on a car that age is that the reverse switch on the gearbox is worn out & needs replacing (relatively cheap).
If you want to double check the wiring diagram have a search for a copy of BMW WDS, this is the official BMW workshop wiring diagram system.