So, what does one do? Taking the clock out for repair is such a hassle I figure I need to come up with a permanent solution to the contact issue. All the engineering solutions I came up with required significant drain current that one would worry about the battery running down. So I decided to use the contacts to my advantage. Instead of the contacts carrying the rewind solenoid coil current, I used the contacts to trigger a MOSFET that carries the the solenoid current. Generally, MOSFETs have a very low off current and one that I had in my stash is very low, low enough where I could not measure any current.
1 Contacts last a long time.
1. Not stock.
2. Increased complexity.
Here is the before and after circuit:
Modification is as follows: