TV being only 3 years old, I'm assuming a flat screen. A two year old, well maintained battery should be ok. But If I am reading your post correctly, and you are only seeing one third battery power available, it still sounds like lack of battery power (amperage). You sure the charging side of your converter is working properly? If it isn't, that could explain why the TV works fine when plugged into shore power.
I am just wondering if your battery is a good one and doing all it can do. But it's not getting charged back up when plugged in and your TV is giving you the first clue.
What you could do is hook up an external automotive type battery charger up to the RV battery and charge it up. Then see if the TV works properly. That would help narrow down the possibilities of your problem.