Here is a typical layout for the 30A Trailer setup... You have two Power Distribution Panels one for 120VAC Circuit breakers and one for 12VDC Fuses. This is the heart of you system...
Note that all of your basic items like lights, heat, etc will operate from Propane and the Battery. The high end items like A/C, Microwave, etc will only run when connected to shore power...
Google image...
Note you can add more batteries if you are going to operate off-grid a lot and then using an Optional POWER INVERTER you can use some of the high end items when camping off-grid...
In regards to your furnace not working it should work jus fine running on propane with the fan running on battery. It has safety features built-in to not operate if everything is not right... i.e. if the fan is not working the furnace will not turn on... I always light up the stove top when I first turn on the propane to be sure the propane is flowing properly. The furnace also has a air vane safety switch that will turn off the propane to the furnace if the DC Fan is not working properly...
I would first make sure the propane furnace fan is working properly off your batteries before turning on the propane bottles...
Most of our camping is OFF-GRID so I survive on a good battery bank. I watch my battery bank performance like a hawk... We do just about everything camping off grid with the exception of the Air Conditioner and the high wattage Micrwave unit. I did get one of the 600WATT manual operated microwave unit that can run off my 2KW Generator for a short time...
Our usual mode of camping is to run all the basic things off the batteries and during the 6PM to 11Pm time frame each evening is where my additional batteries come into play to run outside lighting and home entertainment item. I also like to play with Ham Radio...
My setup is designed to get us by the one day/night running off my battery bank and not deplete my batteries below the 50% charge state by 8Am the next morning. Then I will run my 2KW Generator when allowed during breakfast to recharge my batteries back up to their 90% charge state. This takes around three hours of generator run time... Then we are good to go again for the next day/night run off the batteries...
It all in the planning and finally becomes second nature camping off-grid...
Believe me when I say you just can't just show up in the woods and expect everything to work... It will get dark on you rather quickly haha...
My goal is to wake up in the mornings camping Creekside somewhere to a semi-loud bubbling creek drinking my fresh ground and brewed bean coffee and watching the fish jump in the creek... For me it don't get any better than that haha...
Roy Ken