For live births that occur in a hospital, or a state-licensed alternative birth center, as defined in paragraph (4) of subdivision (b) of Section 1204, the administrator of the hospital or center or a representative designated by the administrator in writing may sign the birth certificate certifying the fact of birth instead of the attending physician and surgeon, certified nurse midwife, or principal attendant if the physician and surgeon, certified nurse midwife, or principal attendant is not available to sign the certificate; and shall be responsible for registering the certificate with the local registrar within the time specified in Section 102400.