Don't know the system, but it should be easy enough to trace it with a simple voltmeter. I would suspect the battery connections first. A bad connection could allow the battery to charge and the converter could provide enough power, but when off, the voltage drop across the battery terminal could be too much.
Without shore power, turn something on that will draw a few amps and measure the 12V starting on the battery connection, not the battery terminals. Then compare it to a measurement on the terminals. If there is any diff, remove the connectors, clean and lub and replace. From there, just follow the wiring.