Disk Utility First Aid Fails on Time Machine Disk
Time Machine makes extensive use of hard (as opposed to symbolic) links (for eg "this folder is the same as yesterday's backup").  See eg http://www.baligu.com/pondini/TM/Works.html.

I thought the issue with copying backup history was that many copy tools won't see the hard links, so you end up with multiple copies rather than links and run out of space, hence your recollections about doing a block level copy instead.

But now I look it up, apparently Finder can copy a Backups.backupdb folder correctly.  See eg http://www.baligu.com/pondini/TM/18.html (which is sadly a little out of date), and Apple's current documentation https://support.apple.com/en-gb/HT202380.

The message in your first screen shot says "You can still open or copy files on the disk, but you can't save changes to files on the disk". So perhaps you will be able to follow the instructions above to copy your Backups.backupdb folder (containing your backup history) to a different drive.