THE ISSUE IS CARRY UP:
Frankly, based on your assessment and comments, I believe Pat Sweeney's "Gunsmithing Pistols & Revolvers", 2nd Ed.(2004), pp.219-220 is the most sensible, a perfectly good solution, AND what I would do, what I have done many times, what S&W factory trained smiths have done in similar situations, and also what members on this forum have done successfully after reading this, which is:
Peening the ratchet tooth (or teeth) to correct timing/cyl ‘carry up’ is so simple, but you're the only one to decide if you can manage it.
Replacing and fitting a new hand may fix your problem and may not, but the hand is not likely needed or at fault. It’s the harder part.
The flat surfaces of the teeth facing you when looking at the rear face of the cyl are where to peen. The tooth at about 3 o'clock is the next to be engaged by the hand (when cyl is closed) to advance the chamber at the right of the one at 12 o’clock, into firing position. The cylinder turns counterclockwise so the hand will engage the 'bottom side' of that tooth. The flat surface facing you is where to peen, on the edge right above the bottom side of the tooth. No need to take the gun apart at all. I lay the gun on a padded surface on its right side, muzzle pointing to the left (I’m right handed) with cyl propped open with a rolled up shop cloth.
If you're worried about force to the yoke and frame, I'm afraid that you're envisioning TOO HARD of a tap. Just a very light peen with a small hammer and punch is all that's needed. The ratchet teeth are not hardened! This takes finesse, not force.
And by laying the gun on a padded surface without restraint as I described, it's allowed to move when the punch is tapped with the hammer mitigating any force to the yoke and frame.
You may not even see the metal deform and it can be enough to solve the problem. One light tap with a small light hammer and flat face punch then close the cylinder and try it. If the cylinder doesn't ‘carry up’ or even if it does C/U but still has too much 'looseness' when fully cocked, give the tooth another tap. You can do all six teeth, or just others where there's looseness with the chamber in firing position when the hammer is cocked.
Rough handling/constant double action rapid fire can accelerate the teeth wear but it did not happen overnight, and now you have another 10 years of shooting before it'll need anything more, depending of course on how much you shoot the gun. If you peen too much and the cylinder carries up too far that puts cocking the hammer in a bind or the bolt 'jumps' out of the cyl notch, not a problem, peen the surface that the hand contacts and push it back.
Also for side-to-side cylinder play, make sure the edges of the cylinder notches are not burred out or the cyl bolt can pop out of the notch. Gentle peening of the notch edges can fix that as well. “Tight lock up” is fine, but can be over emphasized. Recognize the cyl needs some play for the bullet to align the chamber to the forcing cone.
If you got have a new part and are comfortable fitting it, that's fine. But I would not spend money on new parts, because one can easily handle doing the following very minor repair.
The hand is made of harder steel than the teeth since it has six times the contacts of each tooth, and C/U is a typical result after many rounds of shooting. That's why I would not install a new hand, it already has the advantage.
Your hand spring (formerly located in the trigger) is probably in your cleaning medium. Look for a short coil spring with two legs sticking out of it. Or look in the vicinity of the route that the trigger traveled between where you pulled the hand and your cleaning container.
