I do know for sure the .32 Magnum cartridge cannot be inserted completely into a .32-20 chamber, as I tried it once. One would probably not want to ream out the .32-20 chamber to allow full insertion of the .32 Mag (or .327) as the base diameter of those cartridges is somewhat less than that of the .32-20, and the case could possibly rupture upon firing. It would seem to me that a new cylinder chambered for the .32 Mag or .327 would be the answer, or else sleeving the existing cylinder's chambers to accept them. I wouldn't do anything on a revolver that was not heat-treated. From your SN, it may not be, but I don't know exactly what you have. Either way, I think it would be far more costly to do than it's worth. You can load the .32-20 to much higher performance levels than either, if you revolver will stand it. A Colt OP in .32-20 would handle an extremely hot .32-20 load, I don't know about any S&W revolvers that would. So maybe you might want to look for a .32-20 Colt OP instead?
I have a Ruger Blackhawk in .30 Carbine, and had always thought it would be neat to have a second cylinder chambered for .32 Mag. But I never expect to do it, as I can load the .30 Carbine case to any ballistic level matching the .32 Mag or .327 (and even far beyond), so it's pointless.