We have mostly fixed the issue by doing two main things.
- Sliding the drive sprocket further back on the drive shaft. Looking at it more closely (and very hard to see in the pictures) revealed that the point in the chain that was mounted to the elevator was directly above the middle of the drive shaft, not directly above the end of the drive shaft. I'm guessing that over time the shaft collar holding the sprocket in place loosened and inched forward.
- We also removed some of the slack in the steel cable that drives the second stage of the elevator. I don't think this had a big influence, but it definitely reduced some of the shock loading that was going on when the cable was pulled taught.
After these two changes the ratcheting is gone. However, now when the elevator is under a lot of load I definitely see the motor mount bending upwards, thus changing the c-c distance. It doesn't ratchet anymore because the chain is better aligned with the drive sprocket, but it shows us the next thing to be fixed.
Thanks for the help everyone. All of your suggestions have been great.