It would be necessary to build a strong case, not of his mental illness, but of his instability. You will need as much concrete evidence as possible to bring into court.
Even then, the courts generally take the view that until the person harms your child, you have nothing to fear. I'd push for the most stringent (keep your emotions out of it) and hope for the best.
I have full custody of my daughter whose dad is a rapid cycling bi-polar w/ borderline personality disorder. In our orders, he's required to attend counseling and stay on his meds. He's also required to allow the counselor to speak with me. Unfortunately, once he gets done with them, I'm the evil ex and they won't tell me a thing and so far have been very uncooperative. I have no idea if he's still on his meds, but his actions speak that he's off them. The rub is how to prove it when the professionals won't comply with the orders?
I'd recommend going for supervised visitation through a visitation exchange program, if at all possible. Even then, it will be up to the judge and how strongly you're able to present your case reflecting his instability and poor temperment.
I feel for you. My heart is in my throat every time my ex leaves with my daughter. I never know if she'll return or not. Last visit, just this past weekend, he left my home in tears with her in his truck. He was enraged by the end of the visit and was hateful and belligerent to everyone - including the kids. He was physically abusive toward me and my oldest daughter, yet the courts don't see him as a threat to his other kids. Sigh....I don't get it.