How should one deal with a brother who has ostracized his sister because she was blackmailed with exposure of her illicit relationship, even though she has repented and wishes to perform Hajj?

You have done well by repenting from the relationship, and you must be careful not to return to such a thing, and be keen on concealing your past and associating with righteous women. Regarding shunning your brother, strive to mend the matter and seek help from people of virtue. If it is mended, that is what is desired. If he insists, then you have fulfilled your duty and there is no sin upon you. Be keen on maintaining ties with him even if he cuts you off. Muslim narrated that a man said: "O Messenger of Allah, I have relatives with whom I maintain ties, but they cut me off; I treat them well, but they treat me badly; I am forbearing with them, but they are ignorant towards me." He said: "If you are as you say, then it is as if you are feeding them hot ashes, and you will continue to have a˹Divine˺ supporter against them as long as you persist in that."