What is the procedure to give Mother's name only on Child's Birth Certificate?

I am a Hindu woman married to Muslim guy in Oct 2020 during lockdown period. The guy proposed me for marriage and he had insisted me to Vedic Hindu Marriage only as he follows only hindu religion though he is muslim. I had agreed to marry him on the terms that he will not go for second (as multiple marriages is culture in Muslims) marriage in any situation and we will follow only Hindu Religion. After marriage due to lockdown we were not able to apply for our Marriage Certificate till lockdown opened. when we tried for marriage certificate we were came to know that We need to go for Court Marriage under "Special Marriage Act" . I had started the procedure but my husband said that he will do it from his one of the known lawyer. I trusted on him as our marriage life was going well . Within one year in July 2021 , I had given birth to baby boy. I started followup with my husband for marriage certificate and he was only giving excuses and he said every time he will do it on his own. In Dec 2021 , My husband went to his home town at Andrapradesh for few days and came back in jan 2022. After that he has started going to native place frequenly from DEc 2022-jun 2022. I doubted on him and enquired and came to know that he is married with another muslim girl in May 2022. When he came back I asked abt his second marriage. He said that his family forced him and he dont want to continue the second marriage and he will break the marriage within a month. He went on 18 Th jun 2022 and didnt turned up till date. On phone he is only pretending that his Divorce procedure is going on and he will come soon. I don't trust him as this may be a case of Love Jihad. He has also took a amount of worth 80-90 lacs rs during last 2 years from me. Now considering current scenario in the country , It is risky to rely on him and trust him. So i want to continue as a single mother. Luckily becoz of this messed up i didnt apply for my Son's Birth Certificate still though he is 16 months old now. I have enquired abt it but the muncipal officer is saying they will put only his father's name on birth certificate. Can anyone help me in this regards?