What is the jurisprudential ruling on working as a prosecutor in the Public Prosecution, what are its conditions, if any, and is it better to leave it if another source of livelihood is available?

There is no objection to a person working as a public prosecutor, as long as their judicial responsibility does not contradict the rulings of Islamic Sharia (definition). If it does contradict them, such as by prosecuting, holding accountable, litigating, or dealing with cases according to man-made laws that clash with Sharia, then it is not permissible for them to do so, and they must seek another job.