Not trying to be disagreeable or contentious, but .22 rim fire means just that...the firing pin hits the rim of the cartridge. If there is no cartridge in the cylinder, the firing pin is likely to hit the edge of the cylinder/breech face. The impact most likely won't break anything, but could create a 'burr' on the cylinder and cause a bit of difficulty in chambering another round.
Not a large problem with revolvers, but has caused some semi-autos to be less than reliable. Brownells sells a .22 breech face clean up reamer that removes the kicked up 'burrs' at the chamber entrance. I'd imagine that lots of 'smiths have such a tool for semi autos.
Thats not relevant to the K-22. If 'burrs' do occur, you can gently shave them off with a sharp edge. I suggest following OldFuffs suggestion. Another option is after firing the last cylinder, keep the emptys in place if you just gotta dry fire a bit after you get home. You can always partially extract the empty cases and turn them just a bit to present a 'clean' surface for the firing pin to impact. That only works well for about a dozen hammer strikes though...Your mileage will, most likely, differ.
Other comments regarding years of dry-firing with no adverse effects were posted in good faith and are truthful obversations of that person's experience.
Mine are also.
My limited, and personal obversations of dry firing .22s, rests with the notion that a firing pin hitting a hard metal breech face ain't gonna do anything nice to either the firing pin or the breech face.
I dunno, if it works for you, go for it.
salty.