Breadman, the cheese talk certificate of completion is in the mail. It's quite nice. Suitable for hanging
Seriously, though, I'm not certain about the grated cheese, but the 'expensive' galaxy mozzarella, is, imo, most definitely not mozzarella. I would stick to the grated cheese and/or try some other options. Fresh mozzarella (high moisture) (fior de latte, bufala) isn't really suitable for your bake time. Maybe once you get the thicker stone, and trim the bake time a bit further, you might be able to play around with those. In the meantime, though, try to find a brand that comes in both grated and brick versions and buy the brick and grate it yourself. That tends to be the best for this style.
I can't speak for NZ, but here in NY, money doesn't always buy you better mozzarella. The cheapest stuff is frequently the best. At least on the retail end. There's commercial mozzarella that runs circles around the retail stuff. Keep your eye out for mozzarella that's white and firm.
How chewy was this last pie? It looked a bit chewy to me. Don't prevaricate on the gluten flour. Omit it. It tastes horrible and brings no benefit to your crust.
I'm backtracking on the higher gluten flour. If you tell me the beta tends to be chewy, then I believe you. Instead of ramping up the protein, I highly recommend increasing the hydration. Lehmann is 65%, but that's too high for your flour.
Beta + 61% hydration + 2% oil. No gluten flour. With your current setup, it will be a bit paler on the bottom, but it will be a marked improvement over this last pie. Once you get the kiln shelf, this flour/hydration will rock your world
It will be a bit sticky/droopy/hard to work with, but if you want to enter the realm of sublime pizza, you're going to have to get used to it. Use plenty of flour, don't toss it (obviously) and stretch it carefully. The best pizza (and bread) doughs are always a bit difficult to handle.
The impetus to stretch them a bit more is a good one. If you can stretch it to 14, that would really help, both aesthetically and with oven spring. Thinner dough will translate into faster bakes, which is anther added plus- or more color in the same amount of time.
We've all used stones like that at one time another. For what it is, you're getting a lot out of it, but, as I know you're aware, a kiln shelf is a gargantuan leap forward.