Delhi Govt Portal.
Additional copies of the Birth & Death certificate are also issued @ Rs.5/- per copy and on payment of Rs.2/- for search of the single entry in current year and Rs.2/- for every additional year for which the search is made. The certificates are issued from the registration centres/Zonal Offices of the local bodies.
This sounds similar to the process I am familiar with in the US: I have not needed to go to the office in person to obtain a duplicate copy of a birth certificate, but have sent the clerk an application for a copy with payment included, plus a stamped and self-addressed envelope.
If your birth was never registered, then that is a different process, and you would do well to have a lawyer in India deal with that.
Traditionally, there have been alternatives, and allowances for the fact that you can't get copies of nonexistent documents. However, you were born in India, not Syria, so the USCIS official felt it is reasonable to expect that you can obtain a certified copy.
You may want to contact your spouse's congressperson. I found mine very helpful in dealing with a USCIS filing error.
You could also contact an immigration lawyer.
P.S. Search Quora.