i had a couple 12V conveniences in my big truck
the 12V fridge and a 12V > 120V inverter(for my computer, microwave and drip coffee maker)
the truck was equiped w/ "OPTIMUM IDLE" which monitors battery condition and when the state of the battery would decline to near the point that wouldn't start the engine the feature would automatically start the engine and run it at fast idle to restore the charge
both conveniences were battery killers and OPTIMUM IDLE would start the engine from time to time during the night to maintain charge when only the fridge was operating and every time i would use my drip coffee maker or microwave
UNLESS you are plugg'd into shore power or the TV engine is running at or above fast idle i would be cautious about using 12V accessories of any sort specially if the trlr is plugg'd into the TV's system and could discharge the TV's battery to a point that could not start said TV
simple comments based on personal experience
sw