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Unless they have done something new with the furnaces, then it should run off the battery alone as it is all 12V and the only time AC would be brought into the equation would be if the batteries are low and the converter is being required to supply all the 12V for the lights, fridge and furnace. It may not be a problem with the GFCI but simply overloading the 20 amp circuit.
If the furnace cuts out when the GFCI breaker does, then I would say there is a problem with insufficient 12V from the batteries.
Do you get an error message at the fridge when the breaker trips, since it requires 12V as well to operate. Can you operate the microwave with no problems.
I am assuming the GFCI breaker is at the pedestal.
Since there is no "direct" connection between the furnace and AC, I would be looking at the converter and battery condition.
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