For NY style, if your oven is electric and goes up to at least 525, I think you'll save yourself a lot of headaches using that. You probably could get a pizza out of this, but I'm guessing that since you plan on continuing to use it for BBQ, you really can't do major structural modifications with it.
If you really want to give this a shot, I would probably strive for some sort of DIY 2stone setup. What are the dimensions of the shelves and the vertical gap between them?