I think the Onan generators need 12V to generate power, though typically I wouldn't think they'd be wired through the main house 12V disconnect as the generator starter motor takes a pretty large current.
If the batteries are low and the converter is a large unit and the wiring is low impedance, possibly the converter is trying to output more current than the 12V self-resetting circuit breaker is rated for. Possibly the circuit breaker is malfunctioning and kicking out at too low of a current. It would be helpful to locate it and see what the currents are (and what direction they are flowing) when the 12V supply goes dead.
It may also be that there's a fault in the generator circuitry. While a 120V overload could stall the generator, I don't see how that same overload could normally cause the 12V system to shut down unless there was also some other fault.
