Is it permissible to marry the aforementioned girl without her guardian's consent, for the purpose of caring for and protecting her, knowing that her brothers refuse her marriage without a valid reason, and that she desires to marry him?

If a suitor of good character and religious commitment comes to you, her guardians should not prevent her from marrying him without a valid Shar’i (Islamic legal) reason. The guardian is a condition for the validity of the marriage, and if her guardians refuse to marry her to a suitable man without a valid reason, then the woman should refer her case to the Shar’i court so that the judge may marry her. It is not correct to assert definitively that she will be your wife in the Hereafter. If she marries someone other than you, he will be her husband in the Hereafter. If she does not marry, Allah will marry her to whomever He wills, for there is no bachelor in Paradise.