Is it permissible to propose to an engaged woman if her family agrees but she does not, and does the Hadith: "A Muslim should not propose to a woman to whom his brother has already proposed" apply to the suitor in this situation?

Approval of a suitor depends on the girl. If the girl does not approve of the suitor, she is not considered engaged. The prohibition against proposing over an existing proposal does not apply to someone who proposes to her, even if her family approved the first proposal, because she has the right to accept or reject.