OP, when you say your DH has bipolar, has he actually been diagnosed with this? What symptoms of bipolar does he have?
The reason I ask is because his story, word for word is the same as my narcissistic ex. The fact that his ex is 'crazy', she used him as a sperm donor, he's not sure if the child is his. The truth was that he was an abusive irresponsible dickhead, who wouldnt take responsibility for his actions and therefore had to blame someone else (projecting) for his terrible decisions.
The fact that you got pregnant and married so quickly is even more concerning as this is typical narcissistic behaviour from your ex. Lovebombing, fast forwarding the relationship, not taking responsibility for his actions (yes he's paying child support but that's probably only to save public face and he knows he can't get out of it and doesnt want you to know he's really a shit).
As other posters have said, if he really was the great father you claim he is and his ex really is crazy, then what kind of 'good' father would leave their child alone in this situation. If he was a responsible adult, he would fight for contact and establish through DNA that he is the father. The reason he hasnt done this is because he knows he's the father, he probably future faked his ex too and planned this child with her but when she realised what a an arsehole he is, she ran for the hills and refused him contact.
He's lying to you OP and I can bet he's a narcissist. If I were you, I'd contact the ex and get to the bottom of what really happened. Your DC have a half sibling who they could have a great relationship with. Narcissists use the crazy ex card to stop you from contacting them, preventing you from ever finding out the truth.
Don't let this guy mess with your head. You know it doesnt add up. Listen to your gut and protect yourself and your children