Isn't it always a low voltage problem w/ an inverter.
The fact that the line in is controlled thru a wall switch, would suspect the inverter being too far from the batteries, or the wire being too small.
But a volt meter at the tv when turned on will tell the story.
Rich
'01 31' Rexall Vision, Generac 5.5k, 1000 watt Honda, PD 9245 conv, 300 watts Solar, 150 watt inv, 2 Cos 6v batts, ammeters, led voltmeters all over the place, KD/sat, 2 Oly Cat heaters w/ ox, and towing a 2012 Liberty, Lowe bass boat, or a Kawi Mule.