As you have the short certificate, there must be a long form of the certificate registered under the name that appears on this certificate, whether that is a standard birth certificate or an adoption certificate. As it appears you have been able to find no standard birth certificate, it is likely an adoption certificate (essentially a birth certificate in the adopted name). As you know the exact date of birth, you should be able to order the long form adoption certificate from the GRO, entering only the name after adoption and date of birth. If no matching record is found you will receive a full refund. Even though your father was born before 1927, if he was adopted after 1 Jan 1927 it is my understanding he will have an adoption certificate.
Alexander