After re-reading your post again- I have a couple of questions/comments/suggestions. Or maybe I will just, mumble to myself to talk myself through this issue.
Green light "On" on your remote inverter switch....fuse probably not blown on your inverter. I am assuming this led is powered from within the Inverter itself.
Just guessing here but it may still be possible the output of the inverter is zero, and the green light is still controlled to the on state. (Would need to have a schematic on the internals to see if that is possible.) In that case perhaps Inverter died.
I don't know how low the batteries could go before the Inverter stops outputting AC power. Maybe it would not shut off on its own, but I doubt it. But lets say it would ride your batteries to death, I would have to assume the output voltage of the inverter would decrease as well, and after assuming that; your TV might just not be too happy about that and shut itself off. In this case just recharge your batteries.
So going back to square one, I am assuming you did not have shore/generator power at the time of the incident. Because if you did, I would need to be asking you; why are you bringing in AC power to convert to DC power to Invert back to AC power in order to run your TV.
Is it fixed yet? Do you need anymore of my "help"