You bring up a good point. I do use bronze brushes to clean the chambers of my .22's., but don't use them on the rifling. I usually use a .25 caliber or .243 caliber brush. I bend it just reaches the end of the chamber. When I use my pull throughs, I bring them from the muzzle end to the breech.The idea of shooting a different bullet through your 22 to "clean" the lead out will cause more problems than good. The possibility of bulging is there, but something that is much more likely, the regular load that you are shooting through that gun will take a few rounds to "settle" back in to the original point of impact. When I went through several brands with my newest 22 LR, I noticed this to be very true, and depending on the round, groups could be very off of the original POI.
Shoot what your gun likes, clean it as needed, and dont worry about it. I have a 75C that I have owned for over 25 years that I only recently cleaned for the first time a few years ago. Accuracy was never an issue, I only cleaned it because it was having function issues, and I tried that as a starting point to fixing it. Ended up it had a broken spring, may or may not have been a result of lack of cleaning, but the accuracy has always been well within minute of squirrel.