Sounds right.  As you say, the ECU (if it is a model with immo routine such as BEL-era) will check with the cluster to see if the SKC matches what the EEPROM says it should be.  To adapt a new ECU, the EEPROM needs to be patched with correct SKC (MasterJ's tool will reveal the SKC easily, among other methods), or else immo can be manually removed from the original image, or written over with immo-off image (several available on this site) by de-solder or test clip.