I think it is possible that Bae is not the Rumpel’s son. Like he went off to war and she left him because he ran and people died and Rumpel didn’t like that so he took Bae and killed the mother. All of that was hinted at as being possible, but we don’t know if any of it is true. It is just as likely that the mother just left him when she was pregnant with Bae and then Rumpel took Bae from her. It isn’t really clear.
It has been centuries since Rumpel became the dark one so i don’t really think that the fact that it is not common knowledge that Rumpel had a son before he was the dark one means anything. Very few people would have known Rumpel before he was the dark one and most likely everyone would have become aware of him after.