I think I am safe to assume that you have a lathe, or access to one! If you keep it at 0.177, it is just a matter of of copying the two plain external diameters at each end, the overall length, and enough remaining external diameter to fit inside the shroud. The two end diameters are quite critical, the muzzle one to be a close clearance sliding fit in the shroud, the breech one needs to be a close fit into the front of the forcing-cone so that it doesn't rattle.
I decided to ditch the alloy forcing cone and incorporate the shape into the breech end of the barrel, the harder steel seems to slide into the cylinder detents more smoothly.
This might be an easier way to convert to 0.22, it would also leave the original cone intact if you want to switch back to 0.177.