I wouldn't suggest adding to your solar and battery system just to run your fridge, at least not from a cost standpoint. With solar and batteries you are still paying for your electricity, just doing it up front, I'm certain propane would be cheaper than that, especially considering battery replacement costs over time..
Especially if you're getting your propane that cheap, I just got my big residential tank filled and it was over $3 a gallon.
I beg to differ on the "free alternator electrons theory". That would be a violation of conservation of energy, and unless your TV runs on nuclear power that isn't going to happen. The alt has to put out an additional 11 amps which will require additional torque, which will load the TV internal combustion engine which will burn more gasoline....
ICE's are about 18-25% efficient (more basic thermodynamics). I assumed 22%, and 20% alt and wiring losses. That's just the way it is...which is why if you're heating something electricity is not the best way to go compared to directly burning the fuel.
If you really want to run your fridge on electricity cost (and weight) effectively, convert to a 12V Danfoss compressor based system. Those work great and have been around for over 35 years now (I tested and installed my first ones for NASA in solar powered medical fridges in Africa in the early '80's. That was an interesting trip.
It's what most of the blue water cruisers use who either don't want to or can't carry enough propane. The reason this is cost effective is they are heat pumps not absorption coolers and so have a coefficient of performance around 2. That partially makes up for the 20-ish percent engine efficiency, and also would reduce your solar and battery requirements by about a factor of about 2.