How do I obtain a birth certificate if the birth was at home?

Hello, My mother was born in 1955 at her parent's rental home in Mumbai during that time. We needed her birth certificate and had approached the respective BMC ward office for assistance with the same. However, they demand the discharge slip and miscellaneous documentation that a hospital or nursing home provides which we do not have since the birth took place at home. Now our dilemma is that we do not even know whether her birth was ever registered and we have no method to find out since the authorities are demanding hospital documents to search the records and they insist that they will not search the records with just the exact date of birth. We have asked them for any alternate method such as a public record where we could search for her birth registration ourselves but they told us that we have no alternative to getting hospital documents. I need help with the following: Is there a way for me to find out whether my mother's birth was registered or not? In the scenario where her birth was not registered at the time of her birth how do I get it registered now since we are in urgent need of her birth certificate. She currently has her passport, aadhar card, ration card and bank account where her correct name and birth date are mentioned in case they can be used as supporting documents. We have been running around in circles for over a year now. Any advice will be greatly appreciated.