Why you need birth certificate? You have passport and based on date in passport around the world things move.
Even you can approach Indian Embassy in any country and they will give you birth certifcate based on passport date.
You need birth cert. or hospital records? Why you started legal proceedings? Just go to hospital show them school certificates and give them "Gandhi certificate" they will issue you same you can do in local municipality.
Dude you are living in India before leaving learn power of " Gandhi certificate" also you can get easliy from Indian embassy in any country and that remains valid for all purpose including US immigration.
"Gandhi Certificate" ---->> Notes with Gandhi pic.