Install copy Mac OS on b/u external volume

is there a way to install a copy of mac os on an existing back up (external) volume to make it bootable without having to erase and format, thus losing the backup data?

No.

Create a bootable USB installer you can use to boot, install, then migrate from your backup.

Keep the USB updated with the OS on your Mac (you have to remake it).

If you have a T2 Mac, make sure you can boot from an external before you need it:

About Startup Security Utility on a Mac with the Apple T2 Security Chip - Apple Support

If you created it with a cloning utility, it's already bootable. Otherwise, use the Disk Utility to partition the drive.

Working data and backups should be on separate drives; one may fail at any time.

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