The only consumer that should only work with mains hookup connected is the awning light. This is to prevent the light showing when towing, which would be illegal.
I was thinking that the awning light and washroom pump connectors within the BIP unit have been interchanged which would
mean the pump operating only when mains connected and the awning light operating both with and without mains connected.
Other than that I'm stumped and without further investigation can't think what's causing the fault.