This is not really a child custody questions. If she is unwilling to give you power of attorney over the child, the Father would have the first right to have custody over the child. Please consult with a local family law attorney because there are no easy answers here. You could file a third party custody action but you would have to prove that both the Father and the mother are unfit parents if the Father does not agree -- very difficult to accomplish.
Please note that THESE COMMENTS ARE NOT INTENDED AS LEGAL ADVICE and are for informational purposes only. This response is not intended to create any attorney-client relationship and is only based on the limited facts given. The response might change should additional facts be learned and should not be relied on as legal advice. It is recommended that you consult with an attorney who can properly assess the situation, as well as all pertinent facts, prior to taking any action based on the foregoing statements