I don't think that it is possible, but not sure.
The session public keys are stored into NVS to accelerate the session establishing, after first time commissioning.

But I think that even after reading the whole flash into an external file and later flashing it into a new S3, it may not be enough to "clone it".

You can try.
Esptool.py is the necessary tool to dump flash content into a file.

esptool.py --help will print all options.

Another possible way would be by commissioning the backup matter S3 and keeping it stored.

Next time it is turned on, it shall join the network into the Matter environment quickly.