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In addition to raising the stone off the coals you could also use bricks to create a "roof". If you had a second pizza stone supported by bricks, above the original pizza cooking stone, that roof would hold heat and radiate it down on to the top of the pizza. The roof stone could have bricks on top of it to give it thermal mass.
Of course all these bricks would need to be very hot, all pre-heated in the stove well before you begin cooking. But essentially you would be building a mini-brick-oven inside your stove. You would be filling up some of that airspace in the firebox with hot brick, and if the brick was hot enough and close enough to the pizza, it would help speed up and even out the cooking.
This situation is very similar to the first time I tried to cook pizzas on my Big Green Egg. for those not familiar with a BGE, it's a ceramic charcoal grill that looks, well, like a big green egg. The walls are about an inch thick and the thing gets ripping hot. The problem with making pizzas is that your stone can not be exposed to the direct heat of the coals. The first time I made pizza on the Egg, i just put the stone on the metal grill grate and let it rip. Exactly as you experienced, the cheese melted well and the crust puffed up, but it was burnt to heck on the bottom. I checked the BGE instruction manual and it calls for a "plate setter" to be put in, which is essentially a ceramic baffle which prevents the direct heat of the coals from hitting the pizza stone. My next effort included the plate setter and it came out perfectly. You can regulate termperatures perfectly on the Egg which may be quite tricky with a woodstove.