Well-l-l. No. If you're connector is wired to charge the trailer battery from the TV, it's not a fast charge system. Also, at idle, your alternator isn't designed to charge even the car battery. (Up here in NE in the winter you should never "check" you're car to see if it will start in the cold because if it does, it may kill the battery enough not to start when you need it). If the trailer kills your tv battery, you'll need a jump. You'd be better off paralleling a 2nd deep cycle. That will get you 2 to 3 days depending on charge and battery age. If you get a gen, you could even use your fridge while boondocking.
2008 Dodge Durango
2015 Roo 23 ikss