What should a sister, who studies the Quran and has recently started wearing the niqab, do when her mother gives her the choice between staying at home or removing the niqab in order to teach, knowing that she is the only primary teacher for women?

If it is possible to convince the mother for the daughter to go out in her full hijab, that is preferable. However, if the mother insists on the uncovering of the face, it is not obligatory to obey her in that, as obedience is only in what is recognized as good. The daughter can cover her face after she is out of her mother's sight. If the daughter believes that covering the face is obligatory, then she is not permitted to go out except out of necessity if she cannot cover it in front of non-mahram men. But if she believes that uncovering the face is permissible, then there is no blame on her for going out in this state to teach the Qur'an or for any permissible purpose. The daughter must, in all circumstances, honor her mother and treat her with kindness.