If a father refuses to marry off his daughter to a bearded man who adheres to the Sunnah of the Prophet in a non-Muslim country, and the woman has no guardian other than him, then who is her guardian and what can be done to complete the marriage?

The one who will perform the marriage contract for the woman in the event that her guardian withholds consent (ʿaḍl) must ascertain the reason for the guardian's refusal and not be hasty in judging it as 'withholding consent,' for the guardian may have a valid reason for rejecting the suitor. If no valid reason is found, then one must consider the harms that could result from the woman marrying without her guardian's consent, especially in a country where she has no guardians. This is a responsibility upon the Imam of the mosque and similar authorities to be cautious, listen to the guardian, and not merely rely on the words of the woman or the suitor. Some jurists do not rule it as 'withholding consent' unless the refusal is repeated.