Yeah, the script is kinda vague about this part.

Essentially, when you run from Beta Lethera 1, once you re-enter space, STOP. Leave Greyhair docked, let Hero and Ternie defend your mothership from the patrol which shows up (you need to let them get close and turn hostile, don't make the first move to attack). Remember, react, not act

After that encounter Greyhair will talk about fixing the ID. When he mentions "docked" he means HIMSELF docked on the mothership, not the mothership docked at a station. Just wait a while, he'll do it, and the dialog will continue.

I noticed that people who had a problem here were either (1) running somewhere immediately as soon as they enter the system, or (2) had Greyhair out flying. The key is to stay calm, don't lose your head, read the dialog carefully.

Edit:
It's also a good idea to leave game speed at realtime. Only use fast speed when you're travelling; if there's a dialog sequence it's best to let it play out at normal speed.