Pascal if you are able to use legal aid or pay for one appointment with a family lawyer to help you address this and future possible issues, I recommend it. Legal assistance will help anticipate possible issues and try to ensure you get yourself and this child the best outcome.
Based on what you say, you can't refuse to sign the birth certificate if it is so ordered. I'm wondering though if you are certain you are the father, or if there has been a paternity test to establish you as the biological father. If so there is no doubt and you must sign the birth certificate or risk legal consequences. Not sure if that would be a fine or extends to something more serious, but once determined you are the father there is no real way out of it. I agree with other commenters in that the birth mother could be doing this now to seek child support if you are not already paying, the fact she does not let you see your child does not change anything in regards to your child support is calculated independently of access issues and will be based off your income, not anything the birth mother claims you make.
What do you want out of this? Are you seeking to have a relationship with the child, and if so what kind of arrangement are you after? Do you want EOW or 50/50? Since the birth mother is alleging you are the father it will look very bad on her to continue to deny you parenting time and so you can definitely attempt mediation or file for custody orders to set up a parenting schedule which enforces your rights to see your child while holding her accountable if she continues to deny you parenting time after the plan is in place. If the child does not know you you can negotiate an incremental parenting plan where your parenting time increases over a set period of time, which might help ease the child into their new family relationship. You can do all this yourself although utilising legal aid or assistance will make it easier for you in the process, and help protect you. Since the birth mother appears to be initiating a process to get you to pay child support, this may be a good time to work out what kind of relationship you want with this child and get organised to make it happen and put into court orders. It can be a lengthy and frustrating process, especially if the birth mother is combative and high conflict, but make sure you have documented or keep evidence of texts, emails, keep a diary of every time you have requested time with the child, and what the outcome was. If she has let you see the child before, document it. If she denies you ANY visitation, document it. Both of these can help your case in court in different ways. In the meantime, keep your communication with her to the point, polite and relevant. Don't give her any ammunition to try and claim you are hostile, a danger or risk, or abusive.
