How's the cylinder face to forcing cone gap?
I bought a new 586 back in 81 or 82 and it would lock up on me after less than a box of ammo. The gap was about 1/3 of what was recommended (nope, 40 plus years ago, I don't remember the measurements I got with a set of spark plug blade feeler gauges). I fixed it myself with a file and knife sharpening stones. Zero issues since.
Not saying you should fix it yourself, just asking if you can measure that gap and when you call S&W give them the number you get from the measurement.
I can't be getting dirt inside the frame, not from a few shots. it's got to be cylinder to forcing cone clearance or alignment. But if it was alignment I'd think the damage to the bullets would result in lousy groups and some spitting of lead particles to the sides of the revolver when you fire it.
One other issue might be the cases moving back when they fire. Never heard of it on a .22 but on some high pressure rounds (bottlenecked, too) it was an issue.
Can you turn the cylinder with your fingers when it does that? It's crazy how little leverage pulling the trigger really transfers to the cylinder.
Saw a stainless steel S&W revolver lock up (back in the mid 80's again) due to galling of the crane against the cylinder. Lack of lube over the years really screwed it up. On that one (fixed sight .38 or .357, don't remember the model number) you couldn't get the cylinder to turn with even two fingers pulling the trigger. But you could turn the cylinder with your fingers while pulling the trigger. Probably not your issue if it'll go 3 cylinders of ammo before locking up.
Good luck with it. Sorry to hear S&W can't/won't diagnose/repeat the problem when they've got it in their hands.