If you haven't already, get a digital voltmeter. Check your battery's charge every morning. You don't want it to get below 12 volts. (Look up "12 volt battery chart" on goggle.) (That's when you are NOT plugged in to an electric hookup. It will read over 13 volts -- the converter output -- when plugged in even if the battery is almost dead.)
A full charge is 12.7 volts. If running the furnace overnight drops you to 12.0 volts or less then you need to plug into an electric hookup for several hours (overnight, for example). Driving down the road probably will not be enough for a good recharge.