I don't think any family is simple, especially blended families.
We do not know what the lies were, but I'm guessing pretty serious for the OP to be concerned about her career.
At 9, the child is old enough o know consequences, I know at 9, if I told lies I would be punished.
The OP has other younger children to consider and protect, and for her DS not to call the DP Dad, when he's all the child has known, and is in the process of adopting DS is not fair on DS. They are a family, albeit not a 'traditional' one.
We've got a snapshot into someone's life, we don't know the ins and outs, we don't know the poison being spouted from the Childs mother.
From what I have read you seem a pretty concerned, caring SM and mother to your children, maybe the answer is to try to have your DS overnight, maybe have some one to one time with her dad, and you (on seperate days) and be treated as more grown up, maybe going out for the day, shopping, have a pizza, and whatever, maybe ask her to help choose a name for the new baby, or whatever. She might not know where she belongs, especially if her mum has a new partner and has been through a few.
Although she's aware of consequences, she's still a child, probably a very confused one.