I've had lots of professional experience with mentally ill people . I was married to a schizophrenic, also. I don't know what is best for you or your baby. Only a professional person who either knows him or who has enough experience to make wise guesses can adequately advise you. They wouldn't tell you yes or no about involving him in your daughter's life. They would tell you what you need to know about his illness and his personality to make an informed decision.
I recommend that, if he contacts you again, you get his signed permission to talk with his psychiatrist. Or, if you know who the psychiatrist is, tell him what you are concerned about and ask him if he can get permission to talk with you.
Second best would be to talk with another psychiatrist who specializes in psychotic disorders and feels comfortable talking about possibilities.
From your description I strongly suspect that your daughter's father has had a psychotic break which I think would put him into a schizophrenic diagnosis. This is a much more serious illness than schizo affective disorder. With schizophrenia, he may eventually be stabilized on medication which would make being involved in his daughter's life quite doable. The question is, will he agree to remain on medication all of the time. The side effects can be quite disagreeable to the patient. It also takes time to find the right balance of medications.
My ex has been stable and held down a responsible position for 30+ years. He rarely missed work, was highly productive and lived a relatively normal life. He is the exception rather than the rule but many schizophrenics do live reasonably stable lives and are a positive influence in their children's lives. They usually have difficulty with personal relationships, tho we are divorced for reasons other than his illness. I learned that even when someone has schizophrenia they also have their individual personalities that affect relationships.
I strongly urge you to get a better understanding of his prognosis and ways to handle this situation so that it's the most beneficial to your daughter. You will need to tell her at some point that her father is mentally ill so that she can make informed decisions about her health care. I suggest that the more you know about him the better off both you and your daughter will be. That doesn't mean that he has to be physically involved in her life but she does need to know he exists and why he's not in her life when she's older.
I suggest, if deemed reasonable by a psychiatrist, that he be allowed to see her in a supervised setting. She will be able to accept her particular circumstances if she grows up knowing that she will only see him from time to time. I believe that this would be much better than never seeing him and being unable to know for herself what he's like. If she never sees him she's left with too many unknowns and will have questions that haunt her for the rest of her life.
I suggest that you also discuss this with a child psychiatrist that specializes in custody issues. Your situation is just too complicated for the lay person to answer.
Also consider that even tho a court would not grant him parenting time, he still has the ability to petition the court. He can make your life difficult in many ways if you're not reasonable, in his view, of allowing him to see her. Because he's mentally ill, he may pursue action in weird ways. For that reason you may also want to talk with a family law attorney.
Later: You said you want to make this decision now and not wait until she's 2 or 3. This isn't a one time decision unless you decide to disappear and literally deny that he's her father. I urge you to consider the possibility that at some point it will not only be possible but important that she know who he is and is able to see him. Please, please talk with psychiatrists! Especially a child psychiatrist. Your daughter's mental health is too important to make a permanent decision based on the opinions of people who do not know you, her, or her father and his family.
When I read that line, I realized that you perhaps don't know that circumstances happen that require us to be flexible enough to make a different decision at a later time. He is her father. He is permanently a part of her life whether she sees him or not.