Lightroom stores your changes in the catalog, and the only way you can view the changes is via the Lightroom software itself (or you can export the changes to a totally new file, which in no way is considered to be a backup). Lightroom NEVER changes the image portion of your original photos (and thus it would NEVER change the image portion of your backup files either). If you want to use the backup copies occasionally (which is a process I don't really recommend), you still need to use the original catalog, and force it to point to the backup drive.
The idea of using your backup hard drives occasionally as "working copies" will only lead to confusion and possibly changing the backups to "not true backups". I strongly advise you to use your main external drive whenever you use Lightroom.
A complete backup of all your work in Lightroom requires a backup of your catalog file (you haven't really said if you have done this), and a backup of all of your ORIGINAL UNEDITED photos. Thus, for backup purposes, there really isn't a need to have the backups reflect the changes made in Lightroom, nor is there a way to have your backups reflect the changes made in Lightroom.