with 20/20 hindsight, I think I would have used steel framing on the enclosure face with the oven opening and near anything that framed the chimney. In my setup,, near the chimney is the only part of the enclosure that sees any heat, and it isn't much.
Mixing and matching may not be up to building code by you, but I checked with my local bldg office and my only constraints were distances off a building and property lines. Framing in the roof with timber would have been much easier. And wood is easier to buy, cut, and put together too. Maybe concrete board all around or just up front. Plywood would have been OK for sure on 3 of the walls.
So, if everything is very well insulated plus an air gap, I wouldn't expect the wood framing on 3 walls and the roof to see anything higher than typical attic temperatures.
My 2 cents...