Cacher is the code snippet organizer for pro developers

Spoke to Apple Support; apparently it's a known issue. Their advice (which did solve the problem) was:

• Unpair Apple Watch from iPhone
• Sign out of iCloud on Mac
• Sign back in to iCloud on Mac
• Re-pair Watch and restore from backup.

And you really do need to wait until your Watch is absolutely restored, before the option appears in the Security & Privacy screen.

That said... the feature is extremely flaky. 90% of the time, I get the message "Unlocking with Apple Watch" before being presented with the password text box. Pffft.