Welcome to the site. The two guesses are good. Do you have a bidirectional scanner? A bidirectional scanner may help point to the issue. Mine had light, lock, and a parasitic drain at the same time. The issues were traced to the GMIII module. An eBay used module was used as a replacement. All the issues disappeared after cloning.

__________________
'06 X5 3.0i - bought @143,123 miles (12/26/20)