You wont find any real jacketed bullets in the 22lr. The plating on most is simply a thin copper was applied over the lead. I can't speak to the MP-22 specifically, as I do not own one, but I have shot un-plated lead bullets out of a variety of 22 rifles and handguns for many years without any issue. Winchester Wildcats have always worked well, as have various 22lr match loads, most of which are un-plated.

Larry