First of all, there are some handy tips about how to get the birth certificate up on here as a "sticky". I don't know if it mentions the requirements in NY, but it might, so check it out. The requirements are different everywhere, but they usually have discretion right to ask for "additional proof, as needed" whatever that may mean.
What I can tell you is that every human being has the right to citizenship, and that their human rights are violated if they are refused. So if you are still not able to progress, and lawyers are turning you down, you could perhaps turn to some human rights organization? I think that Amnesty International helps refugees on citizenship issues - you could send them an email?
I'm Olivia. I blog about physiological childbirth, homebirth, and unassisted homebirth!