What do you mean 'Wild Camping' or do you actually mean. 'Off Grid'. camping. Which are two different things
Off Grid camping, is basically a campsite where you have no electric hook up, in most cases there won't be any toilet block prevision, only a cold water tap, water waste disposal, a rubbish bin, If you've haven't got your own chemical toilet, they refuse you.
Wild Camping, is basically illegal, as now most councils will have by-laws to prevent unauthorised camping, many years ago you could camp almost anywhere in the New Forrest, find a clearing set up your tent and away you went. Try it now you will at best be moved on, at worst be fined. This includes any overnight stays in the various car parks.
So most people who do Wild Camping, use tents and go way off the main beaten tracks.
But if you're buying from a dealer, I'd would buy from them (often a lot of dealers, will give you a period of time where they will give you a discount on other pieces of camping kit you buy from their store) if you explain that you will be camping off grid, they will provide the right size battery, and will even be happy to pop it in for you.
But do keep in mind, unless you have some way off recharging a battery while away, it will have limitations to how long is will last. So perhaps considering solar power to help to keep it topped up. Another problem with off grid camping, is that not all of the electrical equipment will work. In my caravan, if I'm not using the electric hookup, then I can't use my microwave, and I think if I remember rightly my cooker ignition lighters won't work either, not the one electric solid ring that on the hob.
This will be the same with any television you might purchase for the caravan, if you got a standard domestic caravan, its not going to work on the battery electrics, so you need to ensure that you've got one, that runs both of 12v and the standard 240v.
I see that you have a dog, so a very good quality lead tether is a very good idea, we have t2 dogs, and during the day their water bowl we keep outside but at nighttime, their water bowl is place in the shower tray, so we don't knock it over as we venture to the bathroom, and they don't slobber and dribble all over the floor...