When on shore power and watching TV the source of power is not an issue. Most of the time when I'm off grid, I'm on sat Tv as over the air does not work. I have never seen a sat receiver that works on 12V so having a TV that works on 12v is not going to help much plus I have never seen a big LED TV that has a 12V power converter... has a 12V input. I would not limit my choices of big TVs to those that work on 12v IF you can find one.
Now that I have a home refer with its own power system I keep my sat DVR and TV powered by it so it will record shows and not go through it's 10 minutes powering up cycle.
*I tried that on a small TV that required 16V... it would play the sound but not the picture so make sure it is 12V.