Is this on your Jayco TT? If so the exterior marker lights should be on a total separate circuit powered by tow vehicle. There is probably common ground through the trailer frame. Since the marker lights are led, it would only take a very small amount of stray current.
I would start by checking (cleaning, tightening etc) all ground connections. Then go the positive side of the 12v system, starting with the inverter supply wires.
I'm not sure it's possible for the problem to be on the output side of the inverter, putting stray 120 into the RV causing the 'hot skin' effect.
Ken