If it is working fine now, why worry about it?
Test it in your driveway. It should be working fine. But if it cycles off at 5 minutes, that is a problem. I think it is not related to the inverter use at all. You should have 12 volt power to the furnace with the engine running.
Check the outlet temperature of the furnace. If the air is coming out very hot, it can cause the furnace to shut off on high outlet air temp. The air should be between 95 and 130F. Anything above 130F, check for a blocked vent, and open everything as much as possible.
The problem should duplicate itself in the driveway. At 20F, three electric heaters should keep the RV warm, even without any furnace run time. Run them a little bit, and you would have been fine.
Another possibility is without the inverter set to charge the battery, you might have had low voltage to the battery, and to the furnaces. Your batteries should be fully charged, but if you have not left the RV plugged in while in storage, it will have dead batteries at the start of the trip.
Some inverters require them to be on to charge the battery. It would help to know the inverter type, and if you had it 'off' or 'Charge only' or inverter.
Mine will charge if set to charge only or invert. But will not charge the battery if 'off'.
What does your battery level read? Full? 1/3 empty?
Good luck,
Fred.
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