I cant answer all of your questions but I would say that the inverter is not responsible for charging any batteries. It draws from the battery to convert to 120 v ac. Did you mean CONverter? Is your converter circuit breaker on?
You may want to be careful using the fireplace only as its not forcing warm air into the holding tank area.
I was in the Denver area during the November cold spell. I was burning 30 pounds of propane a day and using a 1500 watt electric heater. I also used fiberboard to build a surround of my holding tank area and had another 1500 watt electric heater under there.
Never again.