It depends somewhat on your television. I have a Samsung 22" (give or take an inch) that draws watts in the low 20s. The Dish receiver draws another 20-some watts. They can be powered by a small inverter, obviously. We had a 150w Samlex which was big enough but it ran from a plug in 12v lighter style outlet. The voltage drop on the 12v side was a continual problem. I switched to a hard wired 300W inverter now.
I bought a power strip that has a switch on each individual outlet. It is plugged into an outlet that is dedicated from the inverter. So no messing around with moving cords and such. When we are on shore power, it works the same as when we are dry camped.
For extra battery savings, you can find TVs that have a 12v brick in the power line, then eliminate that and run it direct off 12v.
Actually our Samsung is that way, so I only have to run the Dish receiver off the inverter. But that is not essential; you can invert and run your TV.
With two batteries (I have the same) you won't want to run the TV for hours and hours. But you can certainly watch it for a couple of hours or so. I am assuming you have LED lights in your MH. If not, do that too, it will give you more watts to play with. Get cheap warm white LEDs on ebay if you need LEDs.