With the rig plugged in, check the voltage at the converter output terminals, and at the battery terminals. They should both be at 13.8 volts.
If there is a difference, the converter is just running into another load, like I said, a weak connection.
Another great thing to have is a voltmeter connected to your battery that you can monitor. You will be able to see the voltage drop when your loads start up, like the fridge and the furnace motor.
One battery should last one night on average with moderate furnace usage, but adding a blanket or two is a better solution to 65 degree furnace temperatures.