In the early 70's I tried to make up some .35 MAS brass (a weird French auto pistol). I chucked up some .357 mag brass (mostly for more length to work with) and spun them on a file to get the head I needed. After trimming and sizing (a lot) they would fit. I loaded cast 121gr round nose 9MM bullets and I think it was unique powder. They functioned, more or less. But it wasn't worth all of the work. I gave that pistol to the Sheriff for display. Black enamel paint and all.
Turned rimmed brass into rimless.
I think I would start with a rimless case and make what ever changes very simple to do. Also, check out the problems, real or not, with the 22 Jet. Never had one but they looked interesting, then with the bad stories, I looked to other toys.