The birth certificate was once mandatory for Passport requirement. Now that Central Government has directed authorities to accept DOB etc., as per qualification certificates, BC lost it's value.
If the full name is registered with authorities at the time of Birth registration, a simple applciation for modification, (with a sworn affidavit duly notarized with supporting copies of certificates) before Registarar of Births is enough for getting revised DOB certificate. There are fees prescribed for such modification.
At the cost of repetition, there is no such additional value for going into such modification, when that is not material in any way.