It sounds like you're picking up (inducing) noise from the inverter, or somewhere.
Are you going to drive around with your RCAs unplugged all the time?
When you have the RCAs connected to something, it cancels out most of the noise. I don't think your problem is really a groundloop. A lot of systems will have 'noise' when the RCAs aren't plugged into something. I think it's some sort of other problem, but i'm not sure what.
So..does it whine when you have the RCAs directly connected to the computer? If so, you might want a GLI to 'isolate' the inverter from the amplifier..