You don't need the steel studs. Think natural building to go with the RMH and there you go, a cob wall with the flue from the heater running through it, or with the barrel in the wall, half in one room and half in the other. Could do it with rammed earth as well. I would not put it in a balloon frame wall for multiple reasons. including the fire hazards inherent in frame construction and the lack of mass to retain heat.
RMH ducting and thermal mass integrated with a wall can be done many ways, but it would be best to avoid setting the barrel into the wall unless you have other ports for cleaning and inspection. You want to be able to remove the barrel in some way and get full access in case core maintenance is needed.
I want my playground back. Here, I'll give you this tiny ad for it:
Switching from electric heat to a rocket mass heater reduces your carbon footprint as much as parking 7 cars