Some LED TV's use a "power brick" that plugs into the wall and converts the electric to 12 volts. If you can find one like that, you can wire it directly to 12 volts. I have a small, cheap, 13" set that works that way. Unfortunately, those kinds of TV's are few and far between.
Using an inverter to power up both the TV and satellite system is an option. You just have to make sure the inverter is rated for the maximum load, and based on the size of the inverter, you have enough batteries to last a reasonable amount of time.
I have heard that it is not too expensive to take a Direct TV receiver from home to use in the trailer, and depending on the receivers you have, there is probably a portable dish that will work - get the make and model number of the receiver you want to use, then check out the dishes you like.
We use a stand alone Dish TV system that uses a Wingard Pathway X@ dish and a 211 series receiver - works great, but have to have shore power .....