I bought an Insignia 24" TV not long ago. Did use it when camper was hooked up to shore power, no issues.
Today I got a 12v cord I've had for a while (came from a car video cassette player that gave up the ghost some time ago) and plugged it into my camper's 12v outlet. Same issue the OP had, some wavy horizontal lines. Probably something I could live with and I also noticed when the volume is up I get a hum which I didn't get with shore power.
I live out in the boonies where there is no over-the-air TV to be had so I couldn't test it with regular antenna TV. I did use my Jensen built-in CD/DVD player and got the aforementioned lines.
So, OP, I'm curious to find out your results.