What a painful and difficult situation for both of you. It certainly sounds like having a child has triggered unresolved grief and pain in your fiance. He is casting about for a way to dull the pain. Alcohol, an affair, and even suicide all look like options to him. Sadly, he is missing out on the baby he has while he mourns the child he lost.
Sometimes people who have lost a child are terrified to love another. They know how vulnerable being a parent makes us. They never want to feel that kind of loss again so don’t let themselves get close.
Some people believe (consciously or unconsciously) that moving on means disrespect for the child they lost. They don’t want “help” because they feel that it would be in some way abandoning the child.
No. I don’t think you are wasting your time to try to support a man who is in this much pain. Your baby is only a month old. Standing by him now may mean that you can yet make a family. If you love him, you need to be patient and you need to get some practical advice. Please consider finding a therapist, not because I think you are ill but because I think this situation would be difficult for anyone. A therapist can give you the support you need and can help you figure out what to do. Once the therapist knows you and your story, she or he can also invite your boyfriend to join you in a few sessions. Sometimes this is a way to show someone that help is actually helpful.
It’s unlikely that you can commit your boyfriend unless he is actively suicidal. It would be far, far better for him to face his pain and accept some help. I hope you can talk with him lovingly and gently. I hope you can let him know that you understand how overwhelmed he is feeling and that you want to be with him in his grief. And I hope the two of you will seek out some counseling to help you through this difficult, difficult time.
I wish you well.