This is for a broilerless oven, but the principles apply to a grill as well.http://www.pizzamaking.com/forum/index.php?topic=21503.msg217026#msg217026
The cookie sheet under the stone (deflection) will definitely help a lot, but, make sure there's a gap between the sheet and the stone- and make sure the cookie sheet is larger than the stone- but smaller than the grill, so heat can pass through the gaps.
I've come up with a new gas grill approach using only aluminum foil. I haven't had the time to write it all down, but here are the cliff notes:
It's basically all the principles of my broilerless setup (link above) but using aluminum foil for the ceiling and the deflector.
Deflector- larger than the stone (smaller than grill), suspended below the grate- you could probably make cuts into the edge of the foil and thread the pieces through the grate so the foil hangs like a canopy below the grates.
Ceiling- the full dimension of the grill
For the ceiling, I'm thinking of tucking heavy duty aluminum into the lid so that it lowers the ceiling to about 5" above the pizza- with a hole punched in the middle to allow the gases to vent.
Another way to do the ceiling would be to fashion something like this out of heavy duty aluminum foil (folded over a few times to give it a little rigidity):http://www.pizzamaking.com/forum/index.php?topic=19861.0
Due to the potential for aluminum to melt above 1100 F, I might be a little reticent to use this in a super powerful grill- I definitely wouldn't use aluminum in a blackstone, for instance, but I think any sub 50K btu grill should be fine- which is pretty much all of them.
With an aluminum ceiling and aluminum deflector, combined with a 600-ish pre-heat for the cordierite stone (use an IR thermometer to take readings), then, with the burners at full blast during the bake, you should see very respectable, balanced fast NY bakes.