First, the SR22 is a 22 caliber SEMI AUTOMATIC, not a revolver. So, you don't have to worry about the finish as much as you would with a revolver. In addition you can remove the barrel from the frame, which means that you can drop the barrel in a cup or glass of solvent. In fact you could drop the barrel in a small ultrasonic cleaner filled with an appropriate solvent for lead.
BTW, those suggestions for using a Chore Bore and wrapping it around a bore brush may work with a larger caliber but IMO will not be practical with a 22 caliber bore. In fact I suspect that trying this with a 22 caliber bore brush will likely lead to a 22 caliber bore brush solidly jammed into your barrel.
Now for a specific solvent for lead which can be found on the Net. First, a warning, the resulting compounds produced by this solvent are HIGHLY TOXIC and mildly absorbable through the skin, so use latex gloves and dispose of any handling materials and remaining solvents in a responsible manner. The recipe for the solvent is 50% white vinegar and 50% peroxide from a pharmacy. The toxic product of the reaction with lead is Lead Oxide, a white powder once used in paint. I also have no idea on the effect of this solvent on the bluing on the outer surface of your barrel but suspect it won't be good, so you may want to think about polishing the outer surface of your barrel for a look that "pimps out" your SR22 a bit.
Usage, best is probably is soaking with intervals spend using a bore brush to expose a fresh layer of lead to the solvent by scrubbing of the oxide products with a bronze bore brush. Second best may be using an ultrasonic cleaner but I've never tried doing this. Time involved can run from 6 to 14 hours, so it's not a fast process. However, I have had to do this with a 357 Magnum and the results were a clean barrel.
Future prevention of a re-occurance. DO NOT USE REMINGTON 22 CALIBER AMMO. Read the Net, Remington's 22 caliber ammunition is not very well thought of. My personal experience with with Remington 22 caliber ammo goes back to the 70's and I wasn't impressed back then and seeing current postings about it's misfiring and leading leads me to continue avoiding it. If you want to shoot lead bullets get some Winchester, the lube used may be smoky but it's excellent for keeping leading to near zero. For copper washed bullets I like Federal, the plating is a bit thicker and it's never leaded in any of my 22 caliber firearms. I also like CCI copper washed when I'm feeling a bit spendy and can't find Federal.