[Penn]

Here's the deal with using a outdoor grill for a pizza oven, by the way your instinct's are correct, but it's a method issue.

We have 4' barrel shaped grill with two temp gauges factory installed on the lid.
The interior has 2 section divided by a thin sheet metal divider, that is removable.
If you have the same remove the divider. The two section are self supporting and each has a bar grill plate 20" X 20" square.

Our method, we removed the divider, replace one grill plate with a 2" X 20"X20" thick piece of smooth slate.

The fire is set on the grill plate side, charcoal and wood. Once started it reaches temps in excess of 650^F with the lid closed, more than enough to heat the stone on the other side. (The fire is built this way to form a natural convection oven. Vents are opened to regulated the flow of heat from one side of the grill to the other, My grill plate side is on the left, fire is built there, circulating the heat left to right. So the vent on the right side are closed, while the right side lid vent is opened, drawing heat across the stone and up.) Sometimes' we place a small pile of charcoal directly under the stone, mindful, that excess heat will crack the stone.

When the fire is right, the pies take about 3 minutes on the stone.

We try to keep the fire at a even burn at all time, but everything about this type of cooking is by feel, unless of course you have brick dome pizza oven in your back yard.