What do you want?
You have to decide just what you want your inverter to power when on battery power. Then you need an inverter that has 120VAC pass-through (not a normal option for off the shelf inverters). At this point you would size the inverter based on first choice (only the TV, then ~ 2 Amp average, about a 500W inverter minimum, P=IE). Inverter would then be installed on that circuit. When on shore power, 120VAC is passed through, on battery power, inverter provides 120VAC (of course it needs to be properly connected to the batteries).
Is best to get a qualified Tech to do this for you. Amateur installs cause fires.