I applaud your ingenuity, but my view is that if you are going to go to all that effort you may as well select the best individual recipes, that is, one for NY style, one for pan style and one for deep-dish style. Of course, pizza operators don't want to do this. They want a "one dough fits all styles" approach. I don't think they worry so much about the type of flour, amount of oil, etc., so long as their customers eat the pies and don't complain. At the end of the day, they will select what best meets their aesthetic and artisanal sensibilities while maximizing profits (hopefully) and retaining customers.
I am not a pizza operator, but if I were this is how I would handle the situation. I would begin by making the amount of dough I think I will need to meet my planned inventory of each style of pizza over whatever time period I normally use. I would scale the weights of dough balls in accordance with the different styles. For example, I know that for a thin 16-inch NY style, I can use around 21 ounces of dough. I would do a similar scaling for the pan and deep-dish doughs based on the sizes of the pan and deep-dish pizzas I plan to make, although proportionately the dough balls will weigh more than the dough balls used for the NY style because of the greater crust thickness for the pan and deep-dish styles. I would coat all the dough balls with oil, put them in corresponding trays, and cross-stack and down-stack them in the cooler. When ready to make the pizzas, I would take the dough balls out of the cooler and let them warm up for an hour or two. For the NY style, I would shape and stretch the corresponding dough balls to the desired size, and dress and bake them. For the pan and deep-dish styles I would roll out the corresponding dough balls or press them into the pans, cover the pans, and let the skins rise for about an hour or so (or whatever time works best for the individual pizzeria). If I had a proofer, so much the better since I would be able to proof the skins at around 95-100 degrees F and with humidity at around 75-80%. Then I would dress and bake the pizzas in the usual manner.
The only difference between the three styles is the manipulation of the dough, not the recipe. If you change your mind and want to alter the product mix, with a little effort you can resize the dough balls, even at the point where pizzas are to be made. Only the manipulation of the dough will change. Of course, in a home setting, you are in charge and can control the processes as you like. Your approach may cut some steps from making two or three separate dough recipes, and it may well be a good compromise recipe to meet three different styles, but it still entails more work and careful planning and orchestration of events than making only one dough to meet the same overall objective. Knowing me, I would mess us somewhere along the way. I have found that I can do a reasonable job of juggling one ball, but not two or three at a time
Please let us know if you decide to pursue the agenda you propose, since I am sure we will all learn a lot from your experience.