My multimeter decided not to work*, so it was all done with trial and error, based on diagrams and schematics for the furnace.
Everything worked on the bench.
But it still didn't work in the RV.
Finally, we traced the wires, through the floor and bulkheads, back to the inverter, to the fuse block, to the battery.
Polarity was reversed.

That RV is a white-ground, black-positive setup. But the factory wire harness changes to RED wire for the negative battery connection, for the last 4 feet of the run. So there are black and red wires running to the battery.
But they don't go where you think they go...

*(I may have accidentally popped the multimeter by having it in "continuity" mode and sticking the probes into a hot 440 VAC outlet. Maybe. No one saw it. You can't prove anything.)