A crewcab dually should provide room for the family especially if you can find one with a bench front seat.
For the dog many folks place the dog in a crate in the 5th wheel/TT and with cameras now days you can watch the dog as you travel. Temperature can be controlled in the 5th wheel/TT either with covered vents/fantastic fans or a generator and the air conditioner.
For our dogs I built a box in the bed just in front of the hitch. It was equipped with vents and a fantastic fan. The box provided plenty of room for our two medium sized dogs.
Cap or Campershell would solve the dog issue when towing the TT until you're ready to move to a 5th wheel.
Although not the most economic option many Rver's travel in two vehicles...Wife and some kids in the family car and husband, kids nd dog in the tow vehicle. It's not bad because it provides a comfortable car to run errands and sight see once you get to your destination. In addition the car or mini-van can be used to carry toys/bikes and water toys.