Is it definitively forbidden for a female teacher to work at a mixed-gender university, wear the prescribed Islamic attire, adhere to the regulations for interacting with students, especially given her need for the job?

The legally recognized hijab is that which fulfills the শরী'আ-prescribed conditions. It is not permissible for a woman to teach in a mixed-gender university due to the fitnah (temptation/strife) and impermissible mixing it causes, and because her presence is considered an endorsement of this evil.

However, if there is an extreme necessity, such as the absence of a provider or a pressing need to work to secure livelihood, then it is permissible for her to work, provided she adheres to the শরী'আ-compliant hijab, maintains extreme modesty in her interactions with men, and continuously seeks work free from mixing.