That does not sound like a ground issue to me. In normal operation, the ground connection should not be part of the circuit at all; it's there for safety in case something electrical goes wrong (like a short circuit to the trailer frame).
I think you have a loose connection somewhere. One place to start is with verifying that all the screws are tight on all the wires in the electric panel. Obviously, you need to disconnect the trailer from power before monkeying around there with a screwdriver!
The power converter is more like a battery charger than anything; it takes 120V power in and produces 12V (nominal) power out. Some models do need a battery connected to work properly. It should operate independently of the microwave.