Hi,
Yes, the furnace fan really goes through power fast.
I doubt that your present battery is a true deep cycle battery. More likely, it is a cross between an automotive (starting) battery and a deep cycle battery.
What I would do would be to:
- Install two (or four) deep cycle (golf cart) batteries in series (or series/parallel if you got four).
- I would also get a catalytic heater such as the Olympian Wave Heater and not bother with the RV furnace at all (keep in mind that this is a non vented heater so you should have an open window at night ... I use a cracked window and fully open vent).
- Look for other places that power is "wasted" such as incandescent light bulbs (replace with LED/s).
- Install solar panel(s) to keep the battery charged.
(note that not all of these items need to be done at once. Do one at a time if needed .... keeping in mind the planned end result so you don't have to tear out anything later)
With such a system, it is very likely that your trailer would never need to be plugged in (leaving today for 10 days of cool weather camping and won't be plugging in at all). However, putting together a system like this isn't cheap.
A lower cost/simple alternative (not the cheapest either, but one that works well for many folks), would be to purchase a quality (and quiet) generator (~$1000) and just recharge your battery as needed. Perhaps a combination of a generator, LED lights (if not already installed) and a good catalytic heater (~$300) would be the best solution for your needs????????
An even lower cost alternative would be to use your truck to recharge the battery (and/or have a second battery that you swap out).
Perhaps the least expensive alternative would be to forego the heater and just wear more clothes.
No doubt you will get plenty of other great suggestions.
Have fun camping.