It almost sounds like the same problem for both the Furnace and the lights. A blown fuse in the battery lead will do the same thing your are describing.
You may have the battery connected up in REVERSE ORDER also. That's about the only thing that will blow that in-line fuse.
Here is what I do when around my POPUP battery. I find the word "NEG" or the symbol "-" on the battery case and make darn sure this is the battery cable that connects to the trailer FRAME GROUND. this should be a BLACK for Positive(+) and WHITE(-) for Negative Frame ground but sometimes they use RED for Positive (+) and Black for frame ground (-). The best to do is look for the word "NEG" and go from there...
Otherwise it will blow the in-line fuse located just close to the battery terminals and breaks the wiring between the battery and the 12VDC fuse panel (Converter/charger unit)
Check for the REVERSE POLARITY first otherwise if it is wrong it will just blow the fuse again when you replace it.
Try that out and tell us what happens
Roy and Carolyn
I claim Horse Creek Country in Southern Ill - Momabear is from North Texas
We live in King George VA
RETIRED DOD DOAF DON CONTRACTOR Electronics Tech 42YRS
"We're burning daylight" - John Wayne
2008 STARCRAFT 14RT OFF-ROAD POPUP with PD9260C and three 85AH 12VDC batteries
2010 F150 FX4 5.4 GAS with 3.73 gears - Super Cab - Towing Package - 2KW Honda EU2000i Gen
K9PHT (since 1957) 146.52Mhz
"We always have a PLAN B"