I'd be interested in learning more about your LEDs and their "resistor". Neither your pic or link are visible.
After you get the lights working correctly the next easy step would be to measure the amp draw. Check the trailer draw, tow draw and both. It's possible that you are at the fuse/circuit limit.
Another area to check is the house battery and circuit. Possibly something is connected to that circuit. I'd hope there is a charge wire and a trailer brake wire.
And since it's also a camper I would expect the house battery to be grounded to the frame including the ground from the tow vehicle. I agree with others that a dedicated ground wire for the lights is best and you may/may not have that.