First off your batteries must be charged up to start with. Then the problem is using the 12v plug in the wall. The wiring from battery to the plug is too small and causes voltage drop to the outlet. The voltage drop under load can be considerable. The 100 watt inverter would most likely work if it were connected with short cables (24 inches) with alligator clamps directly to the battery. The 120v ac extension-cord can be longer. A 100 watt inverter is pushing your luck.
Your television is not receiving enough amps @12v dc to supply the minimum watts @120v ac needed to run the television (brown-out conditions). Too much of that and you will destroy the television circuits and power-block.