My documents/birth certificates are similar to OP's situation, but with less variation in last names. I only submitted the birth certificate from the place I was born (as described on the reciprocity schedule), and then wrote a note on why the name on my passport was slightly different [birth certificate lists me as Anna Barbara Christina Daniella Estella WHITE SMITH (not my real name, obviously, but similar structure with 5 first names), but my passport and my all IDs are only displaying Anna SMITH, and have been for all my life].
Also listed all names on the forms under other names used, when required, and requested my police certificate with all names being checked.
Never shared with them the other birth certificate, since it was not required, and might have created more confusion.