ack don't use the lighter adaptors. They will cause noise since your creating a ground loop. That is multiple grounding points in the car. Just chop off the cig lighter adaptor and run the postivie off your batter then ground your inverter to your frame of the car. Then ground your PSU to that same spot. That should take out 99% of the noise.
If you want to keep the adaptor then ground the PSU to your frame and go to rat shack and try to use a ground loop isolator. They generally suck since they tend to dim the audio .