MacOS – recover partial backup data from a (potentially damaged) Time Machine disk

I am on a MacBook Pro Retina 15-inch, late 2013, with OS X 10.9.5 Mavericks.

First of all, to clarify the question, my goal is to create a new, working Time Machine (TM) disk, that has as much as possible of the old backup data available.

Here are the details of the situation.
I had several years worth TM backups on a 500GB external disk. Few weeks ago I got a message from the system that the disk used for TM backup was about to die. So I used Disk Utility to create a disk image of the TM disk (on another, bigger, external disk).
This disk image creation happened overnight, so I didn't attend the whole process.
In the morning I found:

• the disk image successfully created, with no error messages in Disk Utility,

• the original TM disk completely dead.

(I give this detail because I am not sure of the "state of health" of the disk image, of its file system, etc., and I suspect that the problem could be due to this.)

I have then bought a bigger 1TB external HD, and I am trying to copy the content of the TM saved disk image on it, so to have back a working TM disk, but with no success so far.

I have tried to copy the whole folder, both from the finder, with a (naive) drag and drop, or with rsynch. I have then understood that this type of "file level" method will not work, because of the heavy use of "aliases" (soft-links? hard-links?) used in the TM folder.
So I used Disk Utility to have a "Restore" of the disk image on the new HD.
After some failures, I learned that I hat to uncheck the "Ignore ownership on this volume" box in the info of the target disk, and also temporarily disable TM and Spotlight, to avoid any activity on the disks.
After Disk Utility successfully restored the saved disk image on the new HD, I receive an error about the newly created copy of the TM disk, saying:

Error: Disk Utility can’t repair this disk. Back up as many of your files as possible, reformat the disk, and restore your backed-up files.

I tried many times (as suggested in some posts), but after several attempts Disk Utility still is unable to repair it.

I am assuming that the disk image I managed to save contains damaged data, with maybe the file system messed up in some way.
So, before starting a fresh TM backup on the new empty TM disk, and loose all the old TM backups, my last attempt is to manually copy as much data as I can from the saved TM disk image (which I manage to mount and access via the finder).

So, my question is: what is the best way to copy the folders of a TM backup to a new TM backup disk, one by one, checking each time that the copy was successful, that the TM structure is in good shape, but that the "links" were copied as links, and not as actual files/folders, so to keep the "space efficient" feature of the TM folders structure?

Thanks!