The one thing to be careful about is that 12 volts isn't always 12 volts. The magic power adapter on your computer or TV likely puts out something like 11.5 to 12.5V when it's running.
In your RV, you shut everything down when the battery hits 12.1V. Normal charging in an RV can easily get the power line up to 14.6V. If you have something exotic like T-105's it should get up to 15 or 16V.
So how happy is the television with a 14, 15, or 16V input? Who knows? The guy or gal who designed it was expecting 11.5 to 12.5 volts ....
One alternative - people do make DC/DC converters that will take in the RV battery power and put out a nicely regulated 12V
on the other side.