1. Give it a good cleaning
2. don't use lead (jacketed only) and
3. lube up the rails well.
When new, some of the conversion kits and indeed .22s SIG pistols don't cycle well. Clean off the factory grit and it should work flawlessly within a box of ammo.
SIG did put out some problem mags for the P220 in .22, but those mostly wouldn't seat right in the first place.
Mine (.45 220 with .22 kit) is flawless.
It's the same barrel/slide as the P226, I've put my kit on my 226 as well, with a 226 mag of course.
These kits won't work on the 9mm P220. I haven't studied why, but neither the bottom mag or side mag release P220 9mm models work with the kit. Slide release issue iirc.
Nothing wrong with .22 -- though occasionally (rarely) you could get a FTF from bad primer. Rotate the round and it will go bang. With good ammo this is rare indeed. Yes, you need to clean your .22s and keep them clean. All of my .22 pistols (CZ, SIG, Walther PP, etc.) will do hundreds of rounds in a session without issues, but they get cleaned after each time at the range, and I don't shoot bare lead.
Let us know how you make out!