[Coda]

That's not completely unreasonable. There's logic behind not modifying birth certificates. They're supposed to be documentation that a certain event (a birth) happened at a certain time and place, and certain facts were observed at the time (such as the name given by the parents and the fact that the infant was in fact alive -- and yes, the apparent sex of said infant).

You can't get a birth certificate reissued if you legally change your name, after all. No state I know of will do that.

No, the actual problem is that a birth certificate -- a historical document -- is being required at all. Don't fight against the fact that you can't change your birth certificate when the real issue is that an obsolete document is the only permitted form of documentation. Change THAT rule instead.