I've had similar results when my battery was low.
I have a hand held cigarette socked inverter as well as a larger
corded inverter. In my case, the hand held is less prone to faulting out.
IE- if my battery is low the corded inverter will always fault out due
to low voltage I assume.
The cheapie hand held one will work until my house battery is pretty near dead.
I can only guess there is some small surge when the backlight for the LCD screen
is coming on or various capacitors internal to the TV are charging on power up.
I found my 12v socket was poorly wired. 16 or 18 ga wires on the back and
loose wire nuts.
Forest River electrical is not well done IMO. They often have sloppy wiring.
I suggest a test.
Plug the TV into a good heavy duty extension cord. Plug the extension cord into your
inverter and try connecting the inverter directly to the battery.
If the TV comes on and works OK you can run dedicated, heavy duty wires from your
power center to a nearby 12v outlet installed by you. Be sure and use an inline
fuse or possibly an un-used fused terminal in your power center. I have a couple
spare slots in mine.
We were camping once and my batteries were low and it was bed time
and my wife HAD to see HER TV show. TV would fault the inverter and
refused to come on. I plugged my trailer into my truck, started the
truck, revved the engine while my wife hit the power button on the TV.
It came on. I shut off the truck and she was happy for the duration.
I bought a small genny after that so I can keep my batteries up and
my wife is a happy camper.
Let us know how it works.