Replacing old external hard drive with new backup external drive

If the new drive is an exact copy of all of the photos in the exact same folder hierarchy, all you should have to do is redirect the Lightroom Classic catalog to the same parent (top level) folder(s) on the new drive. This will also relink all sub-folders that are found to be at the same path as on the first drive, which should in turn relink all images in the catalog.

For details, you can read this article by The Lightroom Queen:

How do I move only my photos to another hard drive, leaving the catalog where it is?

You will want to use Option Two because the new drive already has the photos on it.

For example, the picture below shows right-clicking* the parent folder and choosing the Update Folder Location command. The next step would be to select the same parent folder on the new drive, so that this Lightroom Classic catalog will now use that folder on the new drive from this point forward.

*If you haven’t enabled Secondary Click/right-click for the trackpad, mouse, or stylus on your Mac, then Control-click instead.

Not sure I follow you. The current external drive ... is that currently a backup drive? You only use the word backup for the new external drive. In what sense is the new drive a backup?