The roof A/C works on 120v AC. The thermostat and furnace work on 12v DC. Are you sure the house batteries are putting out 12v? It should also work when on shore power, the converter/charger then provides the 12v. Make sure your interior lights, water pump, etc. are getting 12v power.
If all the power works out, next would be checking for voltage at the thermostat. Removing cover should expose the wires that need testing.