Nah, I keep the pie on the screen the whole time, and usually turn in once in the 6-7 mins it takes to cook on the grill.
I throw the pie on when the grill temp gauge hits 620-650deg, normally.
I have a 16" stone that is about 5/8" thick, I don't know what it is made out of, but it is a commercial "pizza" stone.
I also have mexican saltillo tiles, 12" that I custom cut to fit my grill grate, leaving about 3" or so on the left and right side only.
Both the tiles and the stone go right on the grate, no rack. the tiles have cracked into a few pieces, but they easily fit together and they are not getting any worse.. the tiles are thinner at about 3/8 or so.. maybe a little thicker. The stone is exactly 16", so if I use my 16" screen, the sides of the pie get burned sometimes, so I went to the tiles.. I can go full 18" with them.
I cut the biazzo mozz into small cubes, and put them on the pie that way.. you probanly could slice it too, or grate.
I buy a 50# bag of all trumps about 4 times a year on average.. sometimes I get KA Sir Lancelot if I am looking for a change. It works great for bread and pizza.
cooking pizza on the grill is a little tricky, and it took me some time to get it down... i hardly ever have any type of issue at this point.. I estimate I have done over 200 pies on the grill so far. Good luck!
I will take some pics of my grill setup with the stone and tiles for you guys to check out. Oh yeah, I connected my grill to my natural gas line, and drilled out the valves.. so I don't have to keep getting propane.. I can sear the hell out of a steak too. I can get the flames a good 4-6" out of the top of the grates on full-on high.