Hmmmm.... I'm starting to wonder if you might have a meter problem more than a battery
problem. Here's why:
1. 11.9 volts is about 50% of a full charge
2. Most furnaces can't spin the fan fast enough at 50% charge to trip the sail switch and the furnace
does not light
3. The .4 volt drop for overnight with using the furnace
is about right when starting with a full charge, but I would expect to see a lot more drop when starting at a 50% charge.
4. Try connecting your voltmeter to your TV alone and check that reading. It should be 14 volts or more at a fast idle.