You do have til April 15 for state, don't you?
Why don't they just hire someone to call you every day and say, "Did you send it? You're going to send it, aren't you?"
Are they trying to speed up the payment so they can get on with the audit? Or did they decide to audit you when you didn't send the check right away?
I sure wouldn't handle this alone if I were you. Sounds like the deck is going to be stacked against you. I would expect them to play dirty.