The guy did challenge his orders, or refuse to obey anything. He is asking if the status of the ultimate issuer of those orders permits him to issue any orders.
Your reasoning about military orders went out with the Nuremberg Trials. No American may follow an improper order. You cannot legally kill anyone, unless the order to do so was issued by someone constitutionally authorized to order you to do so. Of course, one may have to accept consequences and fight for that right.
As some point or other, MAj. Cook had to present a valid birth certificate. It is not entirely unreasonable to ask the CinC to do the same.
So why don’t they? I would like to see that happen actually. If they can they should squish this guy - the fact that they are ignoring it seems to reinforce Cook’s allegations.