Run a backup from the Dock with no further user intervention
 

The problem solved is to make a one-click backup. Dave Nanian pointed me to an earlier thread in the FAQ, titled
Can I use SuperDuper! 2.0 with my Maxtor OneTouch Button?
which seems to have disappeared.
Here's what I gleaned from that and from some experimenting; this method doesn't require any use of Terminal, FWIW.

• Set up the backup as desired.
• Schedule it. (It doesn't matter what the schedule is because you will shortly delete it.)
• Navigate to ~/Library/Application Support/SuperDuper!/ and copy the .sdsp file (which you just created) from Scheduled Copies to Saved Settings. ("~" means "your home directory".)
• Back in SD!, you may now delete the scheduled backup.
• Using the Finder, control-click on the copied .sdsp file, use Show Package Contents. Make an alias of the Copy Job.app inside the .sdsp package and put that wherever desired, e.g. on the Dock.