[JRTJH]

If the furnace "runs fine on shore power" but not on battery power, the only "changes in power source" is the battery/batteries and the wiring from them to the power center 12 VDC distribution panel with all the fuses. There are two wires from the batteries, one goes to ground the other goes from the batteries through the battery cutoff switch and to the positive lug on the fuse panel.

So, if it works on shore power, the only thing it can be is something from the battery to the fuse panel on that "single wire" or something with the ground "at the battery".