Like many others, I bought a 22" TV and later I discovered it used an adapter to provide 12v that ran the TV. I don't use the adapter and run the TV directly off the 12v supply. I use a small 12v regulator to ensure the TV isn't damaged by the converter output of 14.4v but many others don't use a regulator with no problems
Look around for a TV with an adapter that provides 12v, but read what is marked on the adapter to be sure it is a 12v output, then you will have a 12v TV.
Your 1000w inverter is way overkill. My TV (and regulator) uses only about 1.5A at 12v, or 18 watts.