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.