It's unlikely that the LED tail lights are causing the issue you describe. I would suggest you check the wiring in either the trailer or the adapter - not for corrosion or breaks but check the connections are on the correct pins.

The issue sounds like incorrectly terminated trailer wiring or a short from one circuit to another.

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Wayne
2005 BMW X5 3.0d (b 02/05)
2001 BMW F650GS Dakar (b 06/01)