As others have suggested.. the furnace runs from the 12 volt side of life. If you need to have 13.7 coming into your batteries, like you would get from a charger, and it won't run from 12.7 that you might get from a super good battery, then start the troubleshooting.
My method would be, take your jumper cables, and bring them from the camper battery to near the furnace. Disconnect the negative to the furnace, and connect to the negative jumper. Did that make it run.. then it's the negative feed that is weak. If not.. connect up the negative as normal, and try using the jumper to connect to the positive jumper. If that made it run, the weakness is in the positive feed.
If neither of those help, try connecting the furnace to both jumpers. If that doesn't work, look for a bad furnace part. Something that requires a good voltage source to spark ignition.