Totally correct that hospital "birth certificates" do not work. These are not legal documents, no matter what they look like. It must be the state issued birth certificate. Sorry if my earlier post caused confusion.
I'm an obstetrician...so I'm familiar with the fact that some states many years ago considered there to be only one real "birth certificate," and that was the one that the delivering physician signed and the hospital then sent to the state to "register" the birth. The state then issued a "certificate of birth registration" to the parents. In most states, this was just titled "Certificate of Birth." Then...the feds got into the picture and most states went to the standardized "Certificate of Live Birth" form. Oh what fun terminology can be.