You need to have the agreement enforceable in court. You will need to open a case for allocation of parental responsibilities and then file for relocation of the child. This is a fairly complex area of law and the courts are required to consider 21 different factors when making a decision of relocation of the child. You really need to hire an attorney to get this done properly. To answer your question, based upon the foregoing, it is impossible to say what the domestic courts will rule, based upon the information provided.
Nothing in this communication should be considered legal advice. Nor should this communication lead you to believe that an attorney-client relationship has been created. Only after thorough analysis of your case with you in my office can I provide effective legal advice on your specific issue.