Something is shorting to ground, most likely on the trailer side since you dont seem to have an issue with it NOT plugged in. So assuming you're not using a brake controller or a 7 way, wiring should be as follows.
The brown wire is your culprit as it is all your running lights and what is likely shorting to ground in your cluster or #3 fuse if you will. Best thing to do is unplug the trailer and use a DVOM (digital Volt meter) or an AVOM (analog volt meter) and check the brown wire and anywhere on the frame of the trailer for continuity test. Which if the meter beeps or buzzes [whatever] then its showing a defined connection much like that of testing both ends of a single light bulb.
You will have to either follow the harness and look for a pinched wire or a part of the wire touching the trailer body/frame somewhere.
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