So finally got it working at a consistent clip. Hooray working camera at a discount!
For anyone else trying to remedy an issue with similar symptoms, I would check to make sure that the rack that charges the shutter is making proper contact with the metal piece that disengages the frame-locking pawl mentioned above. It is relatively easy to adjust how the rack sits on the gear that charges the shutter to ensure that it extends fully after each frame. In my case it was not. It did take a little bit of trial and error to figure out which placement worked so that I could fully advance the advance lever, get the shutter to fire, and engage that third piece.
The other thing I would check is that the shutter release on the leaf shutter itself is not jammed in any way. I noticed that the shutter was relatively hard to trigger, so after removing the front element and the shutter speed dial, I wiggled it and fired it a few times from there, and weirdly enough that seemed to finally remedy the issue.
I did slightly shorten my long spring by maybe 3mm, but that did not immediately solve the problem, so I probably would not do that if I were to need to fix it again. I did see that Ace Hardware stocks a very similar spring that may have worked as a comparable replacement. It was about 1.5" by 1/8" with loops and a similar deflection length. Made by Prime-Line.