Theperfectsky:Because of South Carolina's policy regarding mother's rights, in regard to out-of-wedlock fathers, you have a lot of power in regard to your daughter. It's understandable that you want him to have a relationship with his daughter and to have visitation. At any point, however, you have the power to restrict visitation, without any intervention by a court. (the ball is in your court, so to speak).
Just something to keep in mind. You want to keep your daughter safe. Definitely, don't want him to drink alcohol, and be impaired during a visitation with his daughter. At some point, it may be necessary to demand that he get mental health services and detox off the alcohol abuse.
Managing his mental health should be essential for your ex's career as well. There are things he can do, without getting an official BPD diagnosis. He can go to AA meetings, he could get a prescription for some antidepressants. He can take his own steps to learn skills to manage his own emotional dysregulation. He doesn't have to necessarily have to worry about the stigma of getting a BPD diagnosis. He should worry, however, about being an impaired physician and an impaired father.
I thought I'd post the excerpt below from the referenced SC website below for other readers to view. It's likely hard for many people reading this post to believe that a state could have such a policy in this day and age, regarding father's rights.
Unmarried fathers have no rights to their children in South Carolina and, even if they prove paternity, are still under the thumb of their child’s mother. She must consent to his visitation; if she doesn’t, the onus — legal and financial — is on him to prove his worth. Needless to say, no such requirement is placed on the mother.
After reading this confirmation of what you previously stated, looks like you will have no obligation to provide visitations for either of the 2 fathers. Sounds like you realize that as long as fathers aren't abusive with their child, that it is usually important for each child to have a healthy relationship with their respective father. So, sounds like the ball in totally in your court in regard to visitations by both fathers, since both your son and daughter were born in SC (correct?)