Elsewhere 1biggun mentioned his idea for a wildcat on a blown-out 7.62x25 case. I suggested using a .223 case:
There's a thread on Accurate Reloading about a ".357 Maximum Rimless." Actually, there are several if you use their search function; sometimes it's called ".35 Maximum" though.
357 Maximum Rimless with a pic - Topic Powered by Social Strata
I snagged a picture from the thread.
The .223's nominal diameter is only .003" less than the .357 at the base; once-fired brass would be within the usual SAAMI diametral tolerance.
Now, the standard 7.62 AK bolt face will accept a rimmed .357 Maximum perfectly, but you'd have to make up a magazine from scratch, or slice'n'dice a Desert Eagle or Coonan .357 Magnum magazine. The Rimless would need a .223 bolt, but it should feed from a 5.56 or 5.56 mag. Being a straight case, the cartridges should just slide as necessary to line up in the curved magazine.
Numrich has .357 Magnum rifle barrels for $19.95, or did last year anyway.
I don't know if there's a standard straight machine-tool reamer available in the right size, but you can rent reamers. Theoretically you could use a .38 or .357 Magnum reamer, but you'd have to find one that didn't also cut the counterbore for the rim.