Is your current firebox inside the drum/oven? Are you hoping to run the rocket exit through the oven space? I don't believe that would be safe for use indoors if that is what you are talking about, so some clarification is in order.
If you want to run the rocket exhaust around the actual oven space in a sort of double boiler effect, that would be safe and effective.
I would build your rocket burner of brick with fine clay mortar - pure ground fireclay would be ideal, but any decent clay screened to remove anything larger than sand would work. A smooth potter's clay might be easy to get if there is a studio near you. The clay will not act as glue to hold the bricks together, but as a filler/sealer to make sure there are no air leaks, and it will come apart easily and cleanly when it is time to build something else.
Any stone or brick/block that you can stack around your barrel would work for quick easy heat storage. If you build a box to hold it, you can surround a horizontal duct with gravel and rock. This is not as effective as a solid cob mass, but is easily alterable. See Paul Wheaton's pebble bed RMH for an example.
https://permies.com/t/30006/labs/pebble-style-rocket-mass-heater