I believe my rig is basically the same as yours. If that is true, the engine alternator keeps both the engine and house batteries charged. You do not have an inverter.
If you have an LED TV you do not need much power to run it. A small inverter, 100-200watts would run it, depending on the size of the TV. Some of the smaller TVs are 12v and have an adapter to run the TV using 120v ac. Mine does. An LCD TV uses a little more power.
Some good reading to acquaint you with RV 12v power is
The 12 volt side of life.Enjoy your new RV, and lifestyle.
Class C, 2004/5 Four Winds Dutchman Express 28A, Chevy chassis
2010 Subaru Impreza Sedan
Camped in 45 states, 7 Provinces and 1 Territory