All reight big man?
Well, i've tried. And i had a far more powerfull rocket to start with.
First of all, it ain't no rocket if it's not insulated, and in which case, metal is doomed, because it will spall to death.
If i was in your position, i'd use a Batch rocket, because they burn far more than a L or J tube rocket. Send the gases to a black oven, where the smoke goes inside the oven. Insulate the oven (example, fill the space between little barrel and big barrel with rockwool. Even better, fill it with sand, which will keep the heat and act as mass. And insulate outside of the big barrel. And make the chimney plunge in the barrel for a bell effect. Even better, make the heat riser send it's gases under the pizza stone, but keeping a gap between front of teh barrel and the pizza stone, of enough CSA to have the gases enter the inside of the oven via the front, then make the pizza stone short, so the chimney can plunge behind it, lower than the stone, with enough of a gap around to have your right CSA. And put a baffle between the back end of the stone and the bottom of the barrel, so the gases can't go direct from heat riser to chimney.
Well, here's food for thoughts, innit?