Unless you plan to make it a single shot, I don't know how you will get a magazine for it. The original magazine won't work, since neither of those rounds will fit in the clip. You could, I guess, fit a Mauser type magazine, though I don't have any idea what might work, or set a spring cartridge holder into the magazine wall as was done with the German 1888 rifles. It would be a whole lot of work for not much, since the gun is not usually considered among the best for sporterizing, in part because the Mannlicher type receiver makes low scope mounting impossible. Those Italian Carcanos once sold for around $15 (I think Lee Harvey Oswald paid about that for his), and inflation has brought them only up to some $100-150.
Also, while those actions are better than many folks give them credit for, they were designed around a 40k psi pressure level, not the 52k+ of the .22-250. You would also have to open up the bolt face, but that is a minor detail if you get past all the other problems.
Jim