If you are on the verge of exploding, my advice is do it. You're project managing this whole show; it's up to you to decide who calls the shots. Now, I know that response may not be to everyone's liking, but if you let these people dictate terms to you at this point, they will do it all the time. If that suits you, fine, but if it doesn't put a stop to it now.
If you don't actually bite them, at least tell them firmly that you will get back to them. When is suits you. If they are the sort to white-ant you for it you may bear in mind that they would be likely to undermine you whatever you do so stop trying to please them. If they are decent people who are just a bit miffed, it's their job to get over it not yours.