You are right - 'had been' usually denotes past perfect continuous.
However, it is sometimes used in place of 'was', when referring to a single event (in this case a birth) which cannot be continuing into the present.
"He'd been born with that harelip" means the same as "He was born with that harelip".
The inclusion of the word "just" probably doesn't help - "just been" can mean something has happened only moments ago. For example, you could say a newborn baby had "just been born". However, another usage of 'just' is to mean "nothing more than", and in your example, I would guess that the wider context of the quote involved some talk over why someone's lip is the way it is now and they are in effect saying that he was born that way, and the answer is nothing more complicated than that.