Jon, for this particular oven setup, I don't think you're ever going to hit Neapolitan bake times. That being the case, I think you should embrace NY style wholeheartedly. Start off by losing the Caputo and the fresh mozzarella and going with a brick mozzarella that you grate yourself. 650 isn't bad for a hearth temp, but you might want to push it a little higher, if possible.
One of the big downsides to firebrick is that it sucks up a load of heat and takes a very long time to pre-heat. The more mass you have, the worse it gets. For a system with this much energy going into it, you shouldn't be using full size firebricks, but firebrick splits. Even splits will most likely take at least 90 minutes to pre-heat. Charcoal underneath helps, but you have to remember that, with full size bricks, the heat from the charcoal underneath is traveling a really long way, over a long period of time, due to the insulating properites/lack of conductivity of the firebrick.
Getting good undercrust browning in the right amount of time (4ish minutes) is the easy part. To achieve good top browning, you either need a lot hotter ceiling (1000ish), a lower ceiling and/or a ceiling with more thermal mass. Does the kind of grill have the capability to pump vast quantities of heat into the dome during the bake?
Do you keep the top closed during the bake? You might be able to aid top browning a tiny bit by covering the lid with some heat proof insulation, but, generally speaking, thin steel domes of that height are a recipe for failure. A stone ceiling would go a long way in aiding top browning, but the ceiling has to be bigger than the hearth so that it collects the heat coming up from below. Without a second stone, you're going to need a massive amount of heat coming from the side. Does this grill have any kind of forced air option?
Btw, raw flour is very bitter. Generally speaking, you should use one peel for launching and one for retrieving, but, if you only have one peel, clean it carefully after launching so you don't get flour on the finished pie.