Unfortunately, unless it has shipped from Bud's to your FFL, they wouldn't be able to get you the serial number as Bud's wouldn't know what serial number is on the pistol that you ordered until they fill out the FFL paperwork to ship it. I know that's not the answer that you would like to hear. And most likely, you wouldn't be able to get the serial number of the pistol until you picked it up from your FFL.
On a more personal note, I think that it's complete hogwash that NY makes you register any firearms for your permit. I suffered through something similar in Connecticut for a number of years until I moved south.