What can be done about the refusal of the young man's family to his marriage to the girl, due to the age difference and the girl's moderate social behavior, despite the agreement and understanding between the young man and the girl on this matter?

Differences among people are one of Allah's ways. A woman's going out to work is permissible, with conditions, when there is a need. As for the relationship between the questioner and the young man, it is not permissible Islamically. She must end this relationship or transform it into a legitimate marriage. The young man must try to convince his mother to marry this girl. If she is not convinced, then he must obey his mother, cut off the relationship, and repent to Allah.