When using two computers, please copy all database files (at DB and CMDR folder) into the destination folder, not only the Exploration database. The TCE_Exploration.db file is connected to the TCE_Stars.db and TCE_Cartography.db file/database.
The Cargo panel has nothing to do with the clipboard. The star system name is copied into the clipboard as soon as the destination location appears at the Destination display, but will check the auto-exploration specific case.
I just tested it... The Auto Exploration box is updating, but it did not update the clipboard until I activated and hid the Carto panel.