Carolyn,
The main reason most people that dry camp for extended periods (over a couple days) go with the 6 volt GC batteries are as follows:
Usually, the 6v GC batteries offer higher AH than similarly priced 12v deep cycle batteries.
The 6v batteries are made to take the deep discharge and a higher number of discharge/recharging cycles better and seem to last longer in this type of situation.
So for the 3 day, long weekend dry camper, the 2-12v maybe group 27-29 batteries are generally enough and a cheaper option.
For some that like to stretch that to 5 days or longer, the 6v batteries seem to be more favored.
Your situation, having a genny that you can run for a few hours in the nmorning and a few at night, should allow you to do either. With a good three stage charger, it shouldn't take more than that to bring either solution back to 90%-95% charged.
If you decide to go the 12v route, go at least group 27 or 29 to give you the additional AH bank and buy two new ones together. Keep your 24 to power your TV as you described above, it a good plan. Just make sure it's vented if you have to move it off the tongue of your trailer to an enclosed area.