Is it permissible to marry a girl whose family has a bad reputation, despite her good character and the strong objection of one's own family to that?

You erred by establishing a relationship with a foreign woman. Allah has forbidden privacy with a non-mahram woman, commanded lowering the gaze, and prohibited touching her. Hasten to repent and sever your connection with this girl. The fact that you are from the same village does not make what is unlawful permissible, nor does an engagement make what Allah has forbidden permissible. Obedience to one's father in what is good is obligatory, and disobeying him is a great sin. If your father refuses for you to marry her, then it is better for you to leave her, especially given her family's bad reputation. If you desire to please your Lord, then repent from what you have done, obey your father's command, and permanently leave the girl.