The inverter/charger wouldn't charge the batteries since it has no source of 120V power while driving unless the generator is running.
Separately, there's an isolator of some sort that connects the chassis 12V system to the house 12V system when driving to allow the engine alternator to charge the battery. There are a few different designs out there. Many use a high current relay to make the actual connection, and these are a relatively common failure point in the system. (There are other possible failure points, like blown fuses, broken wires, etc.) I would guess that the Magnum Energy panel is not involved in this part of the wiring at all.