It still sounds like you are having a low voltage issue with your batteries. Your furnace is 12v. Plugged in to shore power it is going through your converter providing the 12v power needed. Which has me suspecting a voltage issue from the batteries. Either them holding a charge, connections, ground, but I would suspect its a charging issue with the batteries. How are you ensuring they are fully charged? Have you tried a hydrometer?
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2016 Alpine 3601RS
6500 Onan
2016 Chevrolet 2500 LTZ/Z71
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