I had simillar problems. I realized one day, after I brought the unit inside and pluged it into AC power, everything was fine.
It had something to do with the inverter. and/or grounding of the inverter to the car.
Grounding the inverter solved my noise problems, but created the same problems your describing.
I trashed the inverter and bought a 12v DC-DC supply from keypower. Now all is well.