How can I back up my full iCloud photo library to an external SSD?

Use a Photos library on the external SSD rather than trying to squeeze the originals onto the Mac's internal drive. Copy your existing Photos library to the SSD, then open the copy in Photos. In Photos settings, make it the System Photo Library, enable iCloud Photos, and choose Download Originals to this Mac. That setting downloads into the library, even when the library is on an external drive.

Keep the drive connected while the download runs and check the sync status in Photos. The drive needs enough free space for the full library, not just the size of the optimized copy. The library also needs to be on a suitably formatted drive, such as APFS or Mac OS Extended (Journaled).

When Photos reports that syncing is complete, close the app and back up the external library to a second drive. Open the copied library and check a variety of older photos and videos before considering any deletion.

You can also request a copy of your data through Apple's privacy service. That may be worth considering if you want the files outside a Photos library, though a large request can take time and may arrive in multiple downloads.

Check the resulting files and folders carefully. A download of individual files is not necessarily the same thing as a restorable Photos library with all its organization and edits intact.

Don't use deletion as the test that the backup worked. iCloud Photos syncs deletions, so clearing the cloud library can also remove items from connected devices.

Make a second independent copy first, then open it and spot-check photos and videos from different dates. Compare the item count and library size too, while allowing for differences in how files and library data are counted. If something looks wrong, keep the iCloud library intact until you've sorted it out.