Lights and water pump will still operate on lower DC voltage.......
Furnace.....low on battery voltage and several components will not properly work
Remember it worked fine that weekend for awhile then stopped.........battery had been drawn on (especially furnace which is a DC Hog) and voltage went low enough to cause issues with furnace but not other dc items.
Then you used shore power...AC Voltage converted to DC voltage (converter output 13.2V DC and higher).
Charge up battery.....full charge and try furnace again.