I'd need to see a wiring diagram of the car side to determine what is going on. Some trailer light hookups ( car side ) have separate wires for turn signal and brake lights. If these are bridged I'd expect problems.

Try unhitching the trailer but leaving the wire attached and note results. I wonder if grounding the trailer through the ball if the harness ground is open will cause the problems you have.

I've read that some cars use PWM ( Pulse Width Modulation ) to control brightness of car side lights , I wonder it this extends to trailer wiring.