The reasons for refusal to correct the name in the birth register by the MCD is given below:
Under Section 15 of the Registration of Births and Deaths (RBD) Act, 1969, the Registrar is empowered to make corrections or cancellations only if an entry is proven to be "erroneous in form or substance, or has been fraudulently or improperly made".
MCD considers the birth register to be an authentic historical record of the event as it took place at birth. Since you named your child voluntarily at the time of registration and changed it three months later, MCD views this as a subsequent personal name change rather than a clerical error made at birth. Standard MCD administrative protocol dictates appending an alias (e.g., [Old Name] alias [New Name]) rather than purging the original record entirely.
You may please note that the passport officials generally don't entertain name discrepancies, but under Ministry of External Affairs (MEA) Passport Rules, a Gazette Notification is the ultimate proof of a legal name change in India. MCD does not necessarily need to re-issue the birth certificate with only the new name if you present the full documentary evidences in this regard.
If front-desk executive staff reject the application, book a Public Inquiry / RPO Appointment at the Regional Passport Office (Herald House, ITO or RPO Delhi). Senior officers are well aware that a Gazette notification legally overrides the old name on the birth certificate.
If you still insist on the birth certificate to be issued with the new name alone by the MCD then you may have to file a suit for declaration and mandatory injunction against MCD to declare the new name as legally valid and to direct the authorities to reissue a new birth certificate incorporating the new name.
Once the Civil Court issues a decree, MCD is legally bound to comply and issue the clean certificate.