With the "Egg" I am still trying to figure out a way to get high temp but keep the stone at a reasonable temp ...
I think your true problem lies is the flour you are using! Flours in the US are not capable of handling high heats like the Caputo 00 is, the Caputo really shines in those super high heat environments, but if you cook at lower temps, they look pale and anemic. Caputo is usually used at a higher hydration percent too, so that helps also.
How about getting the BGE up to temp, and adding the stone later to keep the bottom temps down until you get some Caputo flour???