That indeed is a dilemma you have on your hands, and I find it ridiculous that the mom wouldn't have your name listed on the child's B/C considering your presence in the child's life. You might want to have another discussion with her, letting her know that the child is the one that will loose out eventually if you have no B/C listing you as the father to present to the embassy, you also need it to get the right passport for the child. Do whatever you can to have the conversation with her, that's my suggestion.