Is it permissible to remain in a job that involves organizing mixed-gender educational courses, with the presence of a female teacher who does not adhere to the Islamic dress code, knowing that this mixing is an essential part of the work and cannot be avoided, and it is not always possible to provide committed teachers?

No sane person doubts the danger of mixing students. Some disbelievers have recognized this and called for segregation and a retreat from co-educational learning.

It is incumbent upon those in authority to strive to mitigate evil as much as possible. Therefore, we advise you to endeavor to reduce the harm as much as you can. Be keen on separating students by having female students attend the session separately and male students separately. Use a male instructor for males and a female instructor for females, even if it requires increasing the reward or the costs; for safety from sins is unparalleled.

However, if you are overwhelmed and unable to effect change, and you can leave the job without incurring unbearable harm, then it is incumbent upon you to take precautions for yourself and distance yourself from engaging in anything that leads to mixing.