I had my 586 start hanging up halfway through the double action stroke some time back. It finally went away, and I never could find the reason. I have a 627 PC, pre lock, 8 shot that will hang up halfway through the DA cycle also, have not found the reason yet either, although I suspect a close barrel/cylinder gap. The funny thing is that when it locks up, I just rotate it back to the original cylinder notch, and repeat the trigger stroke and it will work fine!
Your main spring strain screw looks a little mushroomed, and that could be enough to cause failures to fire, maybe. Also, .22 ammo is sometimes failure prone on its own, and I shoot a lot of ammo of different kinds before I blame the gun. On a failure to fire, I rotate the cartridge about 180 degrees from the firing pin indent and see it that will fire. Most times it will.
Is your gun a 4 screw or 5 screw? Does it have the wide rib? I have a K2543xx which is a 5 screw wide rib and it is my understanding it is just before the 4 screws began.
Good shooting!!