Hoping to replace entire door handle, lock, and perhaps keeper

Thanks for reaching out to us. Your handle set appears to match our 82-035. This is compatible with the 82-315 key lock, as shown in the video below.

As for the mortise lock, I would recommend the 82-218, but replacement may not be necessary. You can adjust the hook to reach out further (or retract further into the door) by turning the small screw directly above it. Try that first before you buy anything just to be sure.

Finally, for the keeper, see our 89-021D.

That is kind of odd. I wonder if there's some issue with the spring inside the mortise mechanism.

That's easy enough to figure out, fortunately. The first thing we need to do is narrow down exactly where the problem lies. I recommend removing the mortise from the door entirely and then using a flat-head screwdriver to throw it manually. Your goal is to see if you run into the same issue as you just described. 

If you have the same problem, then the mortise is your culprit and it needs to be replaced. Our 82-218 should work perfectly fine. If it seems to work fine, then the issue lies elsewhere; at that point you should examine the thumb lever itself, or even the inside frame of the door to see if there is any damage or obstruction that could be causing an issue.

One thing to keep in mind before you do anything is that your mortise style has a tendency to fall inside the door if you aren't careful. Make sure to extend the hook and keep a close grip on it with pliers or something while you unscrew it.