The furnace needs 12V DC for it's control board and for the blower motor. If DC voltage drops, the furnace won't operate.
The necessary DC voltage can be supplied by either the battery or by the Power Converter. And of course the Power Converter keeps the battery charged when on shore power.
So, if you're on shore power and 1) the DC voltage is too low to run the furnace, but 2) when supplied with 12V DC from the truck, the furnace does work, that suggests that the power converter is not working for some reason.
As suggested, start by checking the circuit breaker in your main breaker panel. It can be hard to tell if a circuit breaker has tripped, so just flip the breaker off and back on. Then check the test panel indicator to see if you've got 4 lights on the battery test.
Also, check for any failures of 120V appliances or outlets in the coach. If some things aren't working while on shore power, you might have a general 120V power problem at the pedestal or connection to the trailer where one leg of 50 amp service is off.
Assuming you have power, the next thing to check is the power converter. I'm not familiar with your floor plan, but the power converter is usually located behind the rear wall of the pass through basement storage.
Check that the power converter is plugged into its outlet. Sometimes the plug can vibrate out of the socket.
Check the on-board fuses on the power converter. If the battery was recently disconnected and reconnected, the fuses may have been blown.
If everything looks ok, plug a drop light or something else into the outlet used by the power converter to verify that it's powered up. Of course you can also use a voltmeter to check for 120V AC at the outlet.
If the power converter has 120V AC at the outlet, and the on-board fuses are ok, take a DC voltage reading on the thick wires coming off the converter. If no voltage, you have a broken converter.