Synaesthesia, thank you for the photos. They really helped me see what I should be doing. I did move my food grate and stone up to the lower set of wing nuts on the KettlePizza insert. Wow! What a difference that made. I also sorted out the problem, at least for me, with the turning forks from the manufacturer. Before, due to the lower height of the pizza stone, I could only come down from above to turn the pizza. Once I had the pizza slightly above the entrance of the insert, I could turn it easily, so long as I turned the upside down, with the prongs pointing up. I used the thin length to slide under the pizza to keep it from sticking to the stone. I may have got one or two small tears, but nothing like before. Keeping it turning was not a problem.
I used a double load of charcoal, and hit my 700F temps without a problem, actually a bit higher, the stone in the high 670's most of the time. I did 3 pizzas and could have done one, maybe two more without reloading. The homemade mozzarella melted nicely, even though it was not as stretchy when we made it as normal.
All in all, I'm quite pleased that I have a pretty good platform to cook on. I can start working on a better dough, improving my sauce, and so forth. But turning out a pizza in 3 to 4 minutes ... easy now. That is such a major improvement over my home oven. It is not a "easy" stick in the stove solution, so if you are looking for convenience along with excellence, the KettlePizza approach is probably not a great choice. You have to get charcoal going, add wood, balance your pizzas in and out of the stove quickly. You have to be there and on it, not sipping a beer and chatting. But I love that. But if you are willing to pay attention, you've got something to do do 3-4 minute pizzas with and for under $400 bucks total: Weber, KettlePizza insert, stone and all.
Photos to follow shortly. Yip! Yip! Yipee!