Is it permissible to take a maid with us for Hajj without a mahram for her, knowing that her travel with us might be her only opportunity to perform the obligation?

One must be cautious about excessive recruitment of domestic helpers, except in cases of extreme necessity. If there is a necessity, the domestic helper must be a Muslim and not a young, beautiful woman, and she must be accompanied by a mahram. If she has a mahram with her, he performs Hajj with her; otherwise, she should not perform Hajj with them. Instead, she should stay with someone trustworthy, unless there is an extreme necessity, and her Hajj is valid.