If I read this correctly, all of the 12V equipment works fine when connected to shore power. In that this unit has some age on it, check the 12V terminations downstream of the battery for loose, broken or corroded terminations. May appear to be fine but can corrosion will act as a high resistance that blocks current flow (including the battery terminals). Often this occurs where a termination block is exposed to the weather. The terminal block where the engine emergency backup starting taps off is a possibility Also check the house battery on-off switch that energizes the 12V is actually conducting closed when in the "on" although, depending how its wired, when "off" even with shore power, the 12V should not function.