Time Machine doesn't believe that a volume is the same as an exact duplicate volume with the same name

• Had a drive named 'External 1."
• It was starting to fail (I had another drive from the same manufacturer, and "External 1" was making the same noises as the other just before it failed.)
• Turned off Time Machine
• Got a new drive, and formatted it, calling it "New External 1"
• Copied everything from "External 1" to "New External 1"
• Unmounted "External 1" and threw it into a box that I keep for hardware failures
• Renamed "New External 1" to "External 1"
• Turned on Time Machine and went to its Preferences. In the list of the things to exclude from backs-ups was "External 1."
• Removed "External 1" from the exclusions list (highlighted its name and hit the minus sign button)
• Checked and found that "External 1" was still on the list of things to exclude
• Repeated several times, but I couldn't get it off the list

Somehow Time Machine doesn't believe that the new "External 1" is the same as the first one (which it did back up with no problems)

What can I do to get Time Machine not to keep "External 1" permanently on its exclusion list, and have it treat it as the same drive as the one I replaced?

By the way, if I rename "External 1" back to "New External 1," Time Machine will back it up. But I can't just keep the "New External 1" name for reasons of scripting, etc.