A small rocket needs frequent feeding because the small amount of
wood that fits in it burns up quickly. You would need some sort of automatic feeder of small uniform fuel to run it overnight. Even then, it is possible for a feeder to malfunction.
I see two issues with your sketch aside from fire duration.
The inner shell you show, if it is metal, will deteriorate quickly if it is well insulated on the outside.
A copper coil wound around the inner shell will be exposed to 1000-1500 F temperatures regularly depending on how closely it hugs the shell, and at that temperature you have an extreme risk of the water in the coil flashing to steam and exploding the coil.