In order to obtain a U.S. passport for a child under the age of 16, both parents/guardians must appear in person with the child and provide consent, authorizing passport issuance to the child. If one parent/guardian is unable to appear in person, then the DS-11 application must be accompanied by a signed, notarized Form DS-3053: Statement of Consent from the non-applying parent/guardian. See https://travel.state.gov/content/passports/en/passports/under-16.html for more info. If your girlfriend's child's father won't cooperate, then she'll need to seek a court order that specifically permits her to obtain the passport without his cooperation.
Out of curiosity, has the father agree to the relocation in writing?