Yes, I was actually going to mention that (I actually started typing then erased it) that maybe the reverse lights don't come on with the trailer or that it might be a coded option but I think you should really focus on the "fast lights" and code, not so much on the reverse light. Trying to check a bus data line signal with a digital volt meter is just going to add to the confusion as well. I don't know if Bdc101 is 100% sure that the module doesn't need to be coded and I have no experience with that.

Double and triple check the basics and your work and try to get a wiring diagram.

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)