I'm not sure if this is the same in the English registration process, but I was getting an "Invalid code" error at first because I was trying to enter the registration code into the Heavensward registration page -- when in fact it need to be entered into the Early Access registration page (they are separate pages on the Japanese Mog Station).

After selecting my service account, there is a big button to register Heavensward, but this is NOT the right button. The Early Access registration button is a smaller button below that, lined up with other options such as hiring more retainers.