The easy way is to pick your 223 based 20 caliber cartridge use off the shelf reamers and dies, then cut the recess for the rim separately.

Brass length becomes the issue.
222 rimmed is hard to come by, even loaded ammo. But there is a ready source of 5.6x50R ammo. Brass sometimes surfaces. The 5.6x50R is 222 magnum length, slightly longer than 204 Ruger. 

An email to any of the reamer manufacturers will likely get you some options for a cartridge based on the 357 max. 
Rimmed versions of 223 based cartridges are popular, using the slightly shorter 357 max is also very common.

First decision would be if you want cheap and available 357 max brass and a custom reamer. Or just shoot short brass. 

Buy 5.6x50 ammo for the brass and use off the shelf dies and a common reamer.