Whats the bore diameter of the barrel? 22 Mag is larger than 22 rim fire.
The smaller barrel is irrelevant. The bullet will swage to fit it. I have shot .357 diam. jacketed hollow points in a .355 barrel for years with extreme accuracy. Lead bullets will conform more easily.
Just don't load a CCI WRF round with a copper jacket at .226 dia.
Amazingly, that would probably still work. It would be interesting to try it and find out. You are not going to blow up a gun with any .22 rimfire round. The worst that could happen would be to get a bullet stuck in the barrel and have to knock it out (easy to do).