I use both those methods and well as using BackupBuddy quite regularly. My best guess is you need to drop the existing connections tables in the destination db before you import the tables from the source db.
re: Unfortunately Connections does not have an export / import functionality, so I am stuck.
Connections does not need such a feature because plugins like BackupBuddy and Duplicator that can transfer site’s data. It would just add to the plugins bulk. It is also better to rely on an import/export that has been put through their paces, such as those plugins. If it were to be built-in, it would use the same process as those plugins and phpMyAdmin. So in this instance it would not help because the same error would occur.
Hope that helps! Let me know how it goes.