Given the size of the wire used for the hot line to the trailer, the TV provides a trickle charge at best. It will top off the battery from the power used when hooking up after disconnecting shore power, but will probably not charge a dead battery up to full (Unless: A--the hot line has been rewired. or B--you drive a real, real long way.)
By the way, I don't like to have shore power and truck power hooked up at the same time. I don't think it can hurt anything, but I would rather not mix electrical systems. Particularly complex ones.
