How can a girl balance her religious commitment and her desire to call to Islam with her family's customs and traditions that do not permit her to wear the legal (Sharia-compliant) attire outside of university?

The questioner must obey her parents, except in disobedience [to God]. She must also be dutiful to them and invite them [to the truth] using available means. As for marriage, if the parents wish to marry her to someone who holds disbelief-inducing beliefs, such as the innovation of the creation of the Qur’an, or the negation of God’s attributes, or the assertion of the distortion of the Qur’an, or the claim that saints know the unseen, or reviling Abu Bakr and A’ishah, or the indwelling of God in His creation, then it is not permissible to accept him. If he holds beliefs that lead to open transgression (fisq) or performs actions that lead to open transgression, then he is not a suitable match for her, and there is a difference of opinion among scholars regarding the validity of such a marriage. However, if he does not hold any of these disbelief-inducing or transgression-inducing beliefs, then there is no impediment to marrying him. It is preferable to marry someone whose religiosity and character she approves of.