If this were true, I would not be saying things I am saying now. Yes they clearly dated in the past as I stated earlier; however, no it was not anything serious. Yes I heard all this from the mother as well. Actually, when I met my fiance I went and spoke with his child mother letting her know I was interested in dating him, since I thought it was only right taking it they had a child together. She informed me that their relationship was well over and that she got pregnant after they broke up when they had a chance encounter. They were never engaged, nor did they ever plan a pregnancy. So from there I took my friendship with him to a relationship further. As we got more serious, the mother of his daughter got jealous and started doing things out of spite like not allowing him his visitation, which lead them to have to take it to court. Which then lead to her requesting child support. Through it all she hasnt worked one day in her life. She chooses to receive help from the state, and from my fiance, while she lives at home with her alcoholic and drug addicted mother. So if I was a person who did things out of spite and was overly involved in things that dont involve myself, I would report her for her actions and the household she lives in since after-all her child and my child are siblings regardless. However, that is not me, I came here for my fiance to get advice from him, and anyone who does not like it can go to a different post.
I think that people would like to help but this one is out of anyone's hands legally and more about how you maneuver. From what you write, it seems that the ex is always about asking for money (what a surprise.) Oddly enough, she gets pregnant after her relationship due to a chance encounter (that should get you thinking, IMHO.) The only thing I'm guessing she understands is the stopping of the flow of money.
To begin, your fiancee/bf would need to get a lawyer in NY since the case is in NY. You can't just file for a child support modification in your state because it's convenient. Second, I don't know how much is being paid and how much of a difference you are talking about. The fact that you've decided to have a child does not in any way diminish your boyfriend's responsibilities, unfortunately.
My guess is that the most effective way to deal the ex is that, if she is receiving assistance she should not be receiving, the potential for her to lose such assistance might be the best way to come to a fair settlement - assuming the current settlement is unfair or in excess of what is adequate for your bf's daughter. It's a tough situation and don't envy the position you're in but it's one that you believe is for the best and good luck to the both of you.