I have been using 1-1/8" OD seamless mild steel tubing. I get the draw-over-mandrel material from Online metals or Speedy metals, also online.
I do use a lathe to cut it to length and round the edges, but careful work with a hacksaw and files would look fine. I don't use an inner sleeve, just make the wood a nice push fit, coat it with epoxy and push it in. That's for the first limb. for the second, after the first epoxy has cured, I coat the inside of second end of the sleeve with wax, then epoxy the second limb in. I do line up the limbs rotationally so that if the wax doesn't let go easily, I can shoot the bow until it does.
I pin the fixed end with a round-headed brad and put a light mark in the other end of the sleeve and the base of that limb with a cold chisel to register the limb when reinstalling it.
Jim