If Hawaii says that their document is proof of birth within their state, then other states are obligated to give “full faith and credit” to that document.
You may perhaps notice that this is subject to the following “Congress may by general laws prescribe the manner in which such acts, records, and proceedings shall be proved, and the effect thereof”.
So no, another state cannot dictate to Hawaii what will constitute proof of birth within that state; but Congress can spell out what would be sufficient or insufficient to establish proof of birth (or any other issue)with any state document or proceedings that every state must give “full faith and credit” to.
Granting your argument about “full faith and credit,” are you quite certain that Hawaii makes no provision for providing a certified copy of the original birth certificate if one is specifically requested?