What is the ruling on having a relationship with a girl who is engaged to someone she does not love and who mistreats her, knowing of her need for me and her desire for me to distance myself from her so that she does not feel wronged towards her fiancé?

It is forbidden for you to propose to this girl as long as she is engaged, because the Prophet (peace and blessings be upon him) said: "A believer is a brother to a believer, so it is not permissible for a believer to buy over the sale of his brother, nor to propose over the proposal of his brother until he gives it up." You must cut off your relationship with her because she is a non-mahram. As for the fiancé's bad behavior, the matter rests with the girl and her guardian. She can consult her guardian and inform him of her unwillingness to marry him, and her guardian has no right to force her. She also has the right to break off the engagement. If the girl breaks off her engagement, then you can propose to her at that time in a lawful manner. Everyone must fear Allah, for engagement is not marriage and does not permit unlawful meetings.