Is it permissible for a girl whose father refuses to marry her to a religious man to appoint a guardian other than the judge to عقد (solemnize) her marriage, especially if she is a virgin?

A guardian should not reject a suitor who adheres to the law of Allah. Rather, he must choose for his daughter a righteous man with whom it is highly probable she will live in happiness. Otherwise, he would be betraying the trust and being deceitful. The Messenger of Allah, peace and blessings be upon him, said: "There is no servant whom Allah entrusts with a flock, and he dies on the day he dies while he was deceitful to his flock, except that Allah forbids Paradise for him."

Efforts can be made to persuade the guardian by seeking the help of virtuous and righteous people. If he responds, then all is well. If he does not respond, then the daughter has the right to refer her matter to the Sharia judge to either marry her off or order her further guardian to marry her off. As for you, suitor, if your marriage to her is not facilitated, then seek someone else and do not despair, for Allah the Almighty said: "But perhaps you hate a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not."