Maddios Pizza chain, at least the one near me, pre-stretches their dough on pans and then covers the dough with tin foil. The pans are placed in racks. When you order, the foil is removed, the pizza is topped, and then the pan is placed into a conveyer oven, so the pizza cooks on the same pan on which the dough was pre-stretched.
Since you are cooking on stone, you will need a way to keep the pre-stretched dough from sticking to whatever will be under or over the dough while you are storing the dough (parchment paper?). As Craig suggests above, opening the dough is very quick. There are many good videos on YouTube showing Neapolitan style dough balls at room temperature being opened very quickly, and that is probably going to be your best approach.