Since no NY commercial ovens that I knew of used convection, I've discouraged many people from using this feature over the years, but, I recently got the chance to use a convection oven myself and it blew me away.
It definitely stretches the boundaries of authenticity, but, thanks the circulating air, it both dries out the exterior of the crust faster/promotes crispiness and creates more even browning. On the pies I used it for, the exterior of the crust was a little crispy while the crumb was soft and moist. It was the perfect combination of puffy moist and crisp. I also think convection gave me a little bit better oven spring, although I can't confirm that for certain.
The one potential downside with convection is that it will brown your cheese a bit faster. As long as you keep the bake time down, this, that shouldn't be a problem.
6 minutes with the convection feature on might produce too much cheese browning, but I think you can shrink that time down a bit. I have no doubt that your stone can store more heat than you're pumping into it in 30 minutes. Rather than extending the pre-heat clock, though, I'd just use the convection feature during the pre-heat. Will this oven allow you do 550 + convection? Some ovens will allow 550 with bake, but if using convection, you have to work at 500.
I'm pretty sure that with a more saturated stone, you might be able to shrink your bake time to 5 minutes. With a stone hot enough for a 5 minute bake on the bottom and the convection feature crisping up the rim, I think you'll be very happy with the results.