Does the coach have an intelic controller? it should be in the same compartment as the large cable that is plugged into the external power source. little square black box about 1"X4"X4" has 5 small wires going to it. If it has a good ground, check the voltage drop across the relay that connects the two systems together during a boosted start. That is the relay that will be closed when the inverter is charging the chassis battery.
Place a trouble shooting light across the relay and see if the light will light when the inverter is charging the house battery.