To my understanding, yes, it is the EEPROM that determines this save game locking. The question is which part. I've always assumed it to be the HDD key, but who knows if that's the case...
1) Choose the system that'll provide the "master EEPROM" which'll be used in all systems.
2) Load ConfigMagic up on that box and tell it to save a backup. That'll go on the root of E:, copy it to your computer and name it something distinctive.
3) Repeat for all other consoles.
4) Copy the master EEPROM to all systems.
5) Load ConfigMagic on each box, unlock the HDD if need be then load the new EEPROM over the old.
6) If any given box is softmodded you will NEED to relock the drive straight after loading the new EEPROM or the system will no longer boot.
EEPROMS aren't compatible between console versions, but ConfigMagic'll re-write them to fit... With the exception of the 1.6 EEPROMs. They must only be written to 1.6 consoles, and likewise, 1.0-1.5 EEPROMs must only be written to 1.0-1.5 consoles.
It may be possible to use a hex editor to copy'n'paste the HDD key from the "master EEPROM" into your 1.6 EEPROM, then write that to the console. But I dunno if that'll work; the system might not boot up if you try it. It would still be quite fixable if the box has a mod-chip that can be flashed without booting the console itself, or if you build an EEPROM reader (which also functions as a writer), but you'd still be risking a bit of hair-pulling.
On the other hand, as I mentioned before I'm not even sure if the HDD key is the relevant bit of data here...