M_pizza, does your oven have a broiler? Will the broiler go on during the cleaning cycle? If it does, move the stone as close as possible to it (2-3") and use that to broil the top (with about a 800 deg. preheat for the stone).
Without a broiler, it gets especially difficult. With an oven within an oven setup (using a stone as a ceiling), depending on the bottom burner output, it's possible to do NY style, but reaching the necessary ceiling temps for Neapolitan is close to impossible.
It's a bit of a long shot, but perhaps, with the right material and enough of it, you can build a ceiling with enough thermal mass that it radiates enough energy to give you a quick bake. Dark materials tend to radiate heat better, so I'm leaning towards soapstone slab or possible 3/4" steel plate. If you could get a nice dark red 1" thick quarry tile, that might work as well. Conductivity is less important than color and thickness.
For the hearth, this may be one of those rare times when the $10-ish 3/8" Walmart stone might actually cut it. If the vertical space between the cheap stone is tight enough (<3"), there's a small chance, with a thick, dark ceiling, you might be able to do a Neapolitan bake with both stone and ceiling at 800. You'd need a long pre-heat for that thick ceiling, though. That could be stressful for the oven, though, not to mention drive up your energy bill.
You could pre-heat your stone ceiling to 900 and slide in a light gauge (read; low thermal masss) piece of steel, maybe 1/8" thick, wait for the steel to hit ~750 and launch, but that's only going to work for one pie. In order to do more pies, you'll most likely have to remove the steel from the oven to let it cool a bit.