Spent a bit of time sectioning some cases suitable for the Martini Cadet... nice change from 9.3mm bullets

As you will see below. the .222 Rimmed (original Australian Super brand) has a quite thick web vs the 357 Max... and lastly a .223 case for comparison.

Look at the brass thickness just below the primer cup on the .357 Max
It would seem the .222 Rem has a working pressure of 38,000 CUP vs the .357 Max's ~46,900 CUP and the .223's ~43,500 so the case should handle it

I've heard the .256 Max was quite popular some years ago (basically a .256 Win Mag based on the .357 Max case) which should give a fairly solid improvement over the original round (can't seem to find any ballistics data), in a more compact package that still leaves a good thickness in the barrel tenon. Some say it will approximate the old .25/35...