The real problem is that the E53 trailer lighting is not really set up for LED's...
People have said you can turn off "cold monitoring" in PA Soft (aka BMW Scanner 1.4), but it didn't work for me.

I pull a race car trailer with LED lights, and I was getting the same issue you have...
I recently bought Curt 57004 (7-way to 4-way) adapter which has circuity designed to fix this. I have used it and it works great.

Curt 57003 is a 7-way to 7-way adapter that also solves this issue.