Is it permissible for a woman who is not in dire need of Zakat money to perform the Hajj (Hajj At-Tamattu') using Zakat money she receives from her relatives?

Zakat is only to be given to the eight categories mentioned in the Quran, and it is not permissible to give it to a wealthy person or to someone who is self-sufficient. Therefore, it is not permissible for this woman to take this money, and it will not suffice the giver if he gives it to her while knowing she is not entitled to it. Furthermore, performing Hajj with this money is not permissible because it must come from lawful wealth, and Allah is good and accepts only what is good. However, the Hajj itself would be valid, though sinful if performed. If the woman is poor, Zakat may be given to her on account of her poverty, not specifically for Hajj. If she saves from it and performs Hajj, then there is no objection.