Your chances are exactly the same as anyone else's. You would only need impeccable finances if everyone else is also required to have impeccable finances. The coop Board cannot impose special conditions on you for reasons of your nationality. You are a legal permanent resident of this country and therefore a protected class of person under Fair Housing Laws.
A Green Card holder is not a foreigner, regardless of their nationality. If anybody attempts to treat you as a foreigner, and discriminates against you because of it, you can file a complaint against them under Fair Housing Law. About the only difference between you and a US citizen is that you can't vote in any elections, and you can't be drafted into the military, otherwise you have the same legal rights and responsibilities as anyone else.
I am very clear on this point because I am not only a licensed real estate professional, I am also a Green Card Brit. Although I have not bought a co-op in NYC, I have purchased many other kinds of real estate in NY, both residential and commercial property, as well I represent clients who have relocated from the UK. You can approach this with your head held high and insist that you are judged by the Co-op Board on the same criteria as anyone else. That is your legal right and there are laws with teeth to protect you in this respect. This is one area of life where America is light years ahead of Europe, and why it is a such a great country to live in.
Good luck with your search.