Thank you for sharing these details, but the part about the unique identifier is still not clear from the screenshots.
Can you please share the temporary admin login details, of the website, for further troubleshooting? I'll also need your permission to download a clone/snapshot of the website, in case it needs to be investigated on a different server.
Note: Your next reply will be private and making a complete backup copy is recommended before sharing the access details.
The new plugin provided by WPML (Import export) fixes this issue perfectly. Many thanks for providing the fix. It makes life so much easier :). Great customer service.