The blower in the furnace takes a lot of power and add in the other 12VDC items, that individually do not consumer much but collectively they are another hit to your storage capacity, and by 3AM your batteries are getting pretty low. As best you can recharge as much as possible before turning in for the might.
To help extend my over night battery capacity I run an electric blanket (135 watts per side) off a small invertor that sips power and set the thermostat to a lower over night temperature. I also cover the sleeping area windows with reflective insulation to reduce heat loss. If temps really get low I'll cover all the windows and pull the slides in. If you can't increase your storage capacity you can reduce the heat loss and run time of the furnace.