Is a brother's rejection of his sister's suitor, based on the claim of negative rumors about the suitor's father, considered a legitimate Islamic justification, knowing that the suitor possesses good religious character and morals and is attested to as such? And are children to be held accountable for the sins of their fathers?

Protecting the honor of Muslims is a legitimate Islamic objective, so Islam has forbidden slander and made it one of the grave sins.

Therefore, it is not permissible to say that a person is an adulterer unless they are known to be openly committing adultery. Whoever slanders a chaste person incurs a grave sin and shall be flogged eighty lashes, their testimony shall be rejected, and they shall be deemed morally corrupt unless they bring decisive proof. Allah, the Almighty, says: ﴿And those who accuse chaste women and then do not produce four witnesses – flog them with eighty lashes and never accept their testimony thereafter. And those are the defiantly disobedient, Except for those who repent thereafter and rectify themselves. Indeed, Allah is Forgiving and Merciful﴾ (An-Nur: 4-5), and He says: ﴿Indeed, those who maliciously accuse chaste, unaware [of the accusation], believing women are cursed in this world and the Hereafter; and for them is a great punishment﴾ (An-Nur: 23).

The focus should not be on the suitor's family, but on the suitor himself. If he is pleasing in his religion and character, then the condition of his family does not matter. The Messenger of Allah, peace be upon him, said: "If someone whose religion and character you approve proposes to you, then marry him. If you do not do so, there will be tribulation on earth and widespread corruption" narrated by At-Tirmidhi.

A guardian (wali) is a condition for the validity of marriage. It is not permissible for a woman to marry herself, as the Messenger of Allah, peace be upon him, said: "Any woman who marries without the permission of her guardian, her marriage is invalid, her marriage is invalid. If he has consummated the marriage with her, then she is entitled to the dowry for what he enjoyed of her private parts. If they dispute, then the ruler is the guardian of one who has no guardian" narrated by Abu Dawud, At-Tirmidhi, Ibn Majah, and others, and authenticated by Al-Albani.

If the suitor is religiously and morally compatible, and you wish to marry him, then your brothers have no right to prevent you, otherwise, they would be obstructing your marriage. If they refuse to marry you off without a valid reason, you may refer the matter to the judge.