Bringing an ORIGINAL document to court has consequences that supersede giving a putative copy to a third party -- such as a news organization.
Deceiving a court with a fraudulent document would be a big no no... Yes?
Whether it would "satisfy" birthers, ant-birthers, or the dog catchers of America, is irelivant... it would "satisfy" the law: in that it would be a legally introduced piece of evidence.
Why the Obama administration doesn't do just that is anyone's guess.
STE=Q