If you are looking for cheap. Then you probably do not want to make your own wild cat.
and based on what you want.
Why not just go 22 Hornet?
Heck, (first typed H E double hockey sticks till I re read it and remembered I would get yelled at)
I'll send you 100 to start with. PM me with an address. Be patient it takes me a little while to dig them up or make them.
You then can go 22Hornet Ackley Improved.
It has a rim is already necked down. There are reamers for the ctg. (you do not have to have a new reamer made as well as dies, forming dies and other.) Not to mention all the load testing you will have to do. While 6 grains of green dot is fine for 38 Spl. will it burst in a 224 version? Or are you going to use black powder you made with charcoal, match heads you get from free packs, and Pee? You said cheap.
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Sorry. My family is Scottish and my grandfather had been know to squeeze a quater so hard the Eagle on the back would burp. Heck the Scotts were the ones that made copper wire. An Englishman dropped a penny and two Scotts fought over it.
My suggestion of starting with 256 Win was so you would not have to get from 357 hole to 257 and all the crushed $.06 brass along the way. The .256 may be a buck a peice for a reason.
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Also the same guys that made the 256 may have wanted to go to .224 but had problems and compinsated by using the little 60 grain bullet.
I remember reading an article in the early 80's in Guns and Ammo about a guy that necked 45ACP to all the then current pistol rounds.
Having a 45-44, 45-38/ 45-9, 45-32, 45-25, and a 45-22 and made barrels for his 1911. If I remember he wanted the 22 and did not want to pay for a 22LR conversion kit and ended up spending 6X, help from G&A, on the 1911 project, in barrels dies, springs, reamers, and all than if he had just bought the conversion kit, the other calibers came about as part of the process of getting the brass from 452 to 224.
Bottom line is you may have to spend big bucks to make a CTG out of cheap brass and no matter how much you shoot there may not be a pay back of not using a commercial round.