What you may want is this: 1) A removable "tray" to hold the fire brick that your cooking on. 2) A roof made of the firebrick.
Turn your rack (that your cooking bricks are on) so that the grates run front to rear, instead of left to right. Make an open bottom (rebar) framework and put your cooking bricks on top of the new rack. It still will sit on top of your existing rack. It will be able to slide out to load/unload. Build a "table" (4 legs and an open framwork top). It can fit inside or outside of your existing stack of firebrick that are around the firepot. Cover the top with the firebrick. The roof needs to be very close, maybe 2" above the cooking bricks. You slide the cooking bricks out, load, and slide them back in. You now have a solid roof just above the pizza. Judging from the last crust on that pizza maybe the roof needs to be 2.25" above the cooking stones. The hot air must escape around the cooking bricks, hit the roof bricks and exit toward the front. With a proper preheat, you probably could cook with the grill door open. Did this make any sense?