A: It could be the converter or as Pass42 suggested, could be wired to the chassis battery. This is unlikely unless someone has been playing with the wiring. I can't see the manufacturer doing that.
B: Yes it will shut down.
Without knowing the battery voltages all we can do is guess. Measure voltage at the battery terminals with a digital volt meter or multimeter. Measure both house and chassis battery voltages. Lights will work with pretty low voltage, so house batteries could be too low for the fridge and still run the lights.
Let us know what you find, we'll have more suggestions as to how to fix the problem.
Also let us know what converter (or inverter/charger) you have, and what batteries.