Does marrying the cousin's fiancée, with the knowledge and consent of the family, sever ties of kinship, and what if she threatens suicide if the marriage is not completed?

You must fear Allah and cut off your relationship with this engaged woman, because the Prophet (peace be upon him) forbade proposing to a woman who is already engaged to another Muslim. It is even more critical if the suitor is your cousin, as this leads to cutting off kinship ties.

This woman is a stranger to you, so it is not permissible for you to have an emotional relationship with her.

As for the woman, if she is not desirous of her fiancé, she should consult her guardian and inform him of her unwillingness. Her guardian does not have the right to force her to marry someone she does not want, bearing in mind that suicide is not a solution.