I'll be honest... I can't help you, other than to say that the 3" 317K would not be a consideration, unless it's lite weight was real important.
I started with a new 4" 617 three years ago in September. I cleaned up the trigger and re-sprung it, added a HiViz front sight, as well as added wood grips (Ahrends retro targets in cocobolo.). I also added a DS-10 speedloader and 8 x10 baseplate. Fun! Next, I added a LNIB 5" 63 from a friend's estate. Miserable trigger - but somewhat improved with some work - and springs. Added the wrong length HiViz sight - works great. Finally, after months of waiting, my new 3" 63 arrived before last Thanksgiving. It's trigger was better NIB than the 5" 63 was after all of my work - and has gotten better. The 617 is still the best, trigger-wise. I added S&W 60 Pro wood grips to both 63's... and bought a speedloader and 4 x 8 baseplate for the 8-shot 63's - uber fun!
I have grown quite fond of the newest - the 3" 63. It is quite a plinker. Always remember, folks will tell you how picky .22 rimfire firearms are as to ammo. That pertains to a self feeder - not a bolt action rifle or revolver - there is no action to work. Sure - you can expect different velocities - and POI - with different ammo fired from a revolver. Some, like CCI Stngers or Velocitors, will be harder to load/unload. It's all in how much you want to spend. I opt for the 36gr plated HP Federal 550 packs at WallyWorld - <$16 - and it works fine - never have had a dud. Being inexpensive ammo, I can blame my odd poor marksmanship on 'fliers' - very convenient.
Another reason to like the 3" 63, should I ever have to seriously downsize my minor collection, is that I bought a 3" 60 Pro - and an S&W holster to fit either 3" J-frame. The 4" 617 is fun - but in a carry situation, it starts out 50% heavier than the 3" 63 - and I don't have a holster for it. Seriously, consider either the 4" 617 or 3" 63. Obviously, I couldn't care less whether it has an IL or not.
Here is that new holster - with the 3" 60 Pro in it, and the 3" 63 alongside. It came from S&W Accessories (~$75 with S/T & S/H) - perfect fit. Oddly, I bought two sets of '60 Pro' grips - liked them so well, I bought a 3" 60 Pro in January! I never thought I'd enjoy J-frames... wrong!
Stainz