A tv with an external power supply can be made work by plugging it into the cigarette lighter. I had to get a lighter plug that upped the voltage, 16 iIRC, but it is easy to do and won't have the parasitic draw of an inverter.
A better use for an inverter is the microwave. On my TT I installed a Tripp Lite 2,000 watt inverter/charger. I think it was an APS 2012. The only problem was that the microwave did not like the MSW power and did not work correctly. I would spring for a PSW model. I made mine whole house and fed the panel by use of a 2 way automatic switch for the inverter and the plug. I mounted it in the front storage compartment and ran AC th the rear underneath the panel, I used exterior wire. I had to drill 2 holes in the floor. One in the front to deliver heavy cable to the inverter charger, and one beside the panel for the AC. Be sure and fuse the DC cable before it goes into the inverter/charger.