1. Either toss the potentate and 00 or use them for something else like bread. Neither will perform as well as Full Strength.
2. Bubbliness, aka oven spring, is a factor of proper fermentation and an oven setup that will give you a fast bake time.
Is your dough between 2x and 3x the volume by the time you stretch it?
How long are your pizzas taking to bake?
The most important contributor to puffy NY style pizza is bake time. No amount of formula/flour/fermentation time tweaking will make up for an improper oven setup. You need to stop bouncing around to different flours, formulas and fermentation times and settle in on one flour (Full Strength), one formula (Lehman), and one fermentation time (2 days) while you test your oven and see how quickly it can bake a pizza.
If you absolutely cannot wait 2 days for dough, then trim it to 1, but, if you want any kind of flavor to the crust, it has to ferment at least overnight.
Unless you knead slowly, 10 minutes is probably too much kneading by hand, and probably a bit of a pain in the butt. Mix until the dough comes to a ball and then knead about 2 more minutes. The dough should look somewhere between cottage cheese and smooth when you're finished kneading.