Half ton extended cab pickup (of your preference) with a camper shell would probably suit your needs perfectly. Should be plenty of room in the cab for the family/dog and plenty of dry storage in the bed for whatever else you need to bring, along with extra sleeping quarters if needed in a pinch. Outfit the truck however you see fit, they can be had with quite a lot of 4x4 equipment from the factory, including a selectable rear locker, skid plates, etc. I think with that small of a camper, a 3/4 ton or 1 ton would be overkill, especially given how capable modern half tons are. Hell, a diesel Colorado would probably be capable enough at this point but might be a little cramped in the cab for the whole family.
I have a mid 2000's Chevy silverado with electronic 4x4, it has an auto mode which is great for snow/gravel/light mud and can also be locked in 4hi or 4 low. I didn't get the z71 with all the off-road stuff as I have a Jeep for that, but it does it's job well and has plenty of room for a family of 4 with a decent sized (75lbs) dog in the cab. We've never full-timed in it but it'd mostly just be a matter of a combo washer/dryer in the camper and a few other food prep things vs. what we usually take. The enclosed bed is great for both bringing things with us without having to load it all in the camper (we usually leave the camper packed lightly) and also if we have guests - just add an air mattress in the back of the truck and you have an extra full sized bed. We tow a 5000lbs when loaded camper without issue, so I doubt you'd have any issue with your smaller camper unless you pack REALLY heavily.
I don't know how long you plan to stay in one place, but we camp around once a month so a decent amount of towing (1-3 hours away). 4x4 doesn't get used too much but as always, when you need it, you need it. I've also camped with just the truck by myself or with my wife and it was fun - certainly beat having to set up a tent and much, much better in cold or rain. Plenty of storage in the cab on those trips, but usually we need the seating.
As to Chevy/Ford/Dodge - if I were to buy another, I don't know that I'd get the Chevy. It's pretty good, no major problems but enough minor ones and some worrying corrosion that I might try something different next time. Of course, they've changed a lot since I got mine.