It's not possible for single mummy to add the natural father name to the BC , as the natural father is married. In order to add the father's name you required a Certificate of Marriage.
No benefits will be provided .. as you will be registering as a single mum. Does the natural father want to add his name to the BC? What did the natural father suggest, is he getting a divorce?
To avoid any complications, its the best to register as a single mum.