I think an "undo" pattern is probably a good idea, because you have an original and a duplicate which is identical, so the user has to think "which one to delete?".
You might at first glance think it doesn't matter- it's just a label. But it might have conditional access tied to it in the back-end which would be blown away if the original is deleted (unless of course there's an "undo delete" pattern ;) )
Personally, I think the whole thing needs a re-thinking. Why not click the duplicate button, and via AJAX/JSON Moodle duplicates the label/activity, redraws the page/DOM with the new duplicate and throws a new AJAX notification at the top of the page using 'generalbox' with a success/error message (and a undo/retry button) that can be dismissed or fades after 5 or so seconds - kind of like the "drag and drop" notification at the top of the page when you turn on editing.
but that's just wishful thinking, one thing at a time