With the small Martinis, the issue is size of the cartridge, what can be shoehorned into the breech. In Australia a rimmed .222 was a popular conversion for the .310 Cadet rifles.
There is something else to consider when selecting a cartridge for conversion - the back thrust against the breech. There are a couple of factors here; pressure is one, the area of pressure application is another. This is why the original Thompson Center single shots used high intensity small rounds, but more moderate larger ones.
A .357 Maximum is a relatively high pressure round, but it is rather small in diameter, so the actual backthrust may be less than a larger, more moderate round.
Get out your calculator, look up the pressure of your chosen cartridge, calculate the area of the inside of the base of the cartridge case (which is where the back pressure is applied), and determine how many pounds of backthrust there will be.
Another issue is the firing pin. For a higher pressure round, a small diameter firing pin is necessary. The Brits used a bar dovetailed across the face of the breechblock when they converted the .450s. Nowadays, a bushing threaded into the breechface would be more likely.
Trick is to find a rifle with a crisp action that has already been bubba'd beyond restoration. Come to think of it, I recently saw a sported .303. I might go have a look at it.