since you didn't list what TV you have, it's hard to say about it charging.
but most TV's only provide a trickle charge AT BEST.
if the batteries were real low, it would take hours of driving to recharge them.
as long time dry campers, we also have two deep cycle batteries but never leave without our Honda 2000. we can usually go 4-5 days if we don't use the furnace. but if you're using the furnace a lot, you can drain two batteries in 2 nights of usage.
and Zoe the Wonder Dog(R.I.P.)
'07 Roo 23SS pushing an '07 Chevy Avalanche
Equalizer WDH and Prodigy BC