Hi,
Have you checked to see if the CONVERTER is powered while plugged in? It may have fuses/circuit breakers on it's input and output which can reset. (While you are there, note the make and model of the converter for future reference.) As it charges your house batteries and supplies 12V to the house portion of the vehicle, it would be a good place to start checking, especially since your house batteries seem dead.
Many Generator sets start from the house batteries. If the main engine starts, you could leave it running at a fast idle and operate the emergency start switch while trying to start the generator. (This presumes you have such an arrangement available, most do!) It should work and the generator should, at least, turn over.
The fridge also need 12V, even on propane, to supply its circuit board as do the water heater and the furnace so its not starting is no big surprise.
You could also try a 'portable' voltmeter across the batteries to see what the voltage is remaining, if any, and where it disappears. Anything under 11.5V is considered discharged. When the inverter is operating it should read at least 13.5V across the batteries. Check the liquid in the battery cells at the same time and make sure it has not boiled away. Add distilled water as necessary.
