What is the ruling on working in a governmental postal institution that provides usurious and prohibited commercial services, especially with the presence of mixing (of genders) in the workplace, and is it permissible for an employee to be promoted to a managerial position overseeing these operations, even if he avoids directly engaging in them, but consequently misses congregational prayers?

Dealing with usurious money and commercial insurance is forbidden (haram), and you must repent and leave work in this field. You are permitted to work in permissible areas within the institution, such as collecting and distributing postal mail and packages. Supervising forbidden acts or ordering them is forbidden, even if its prohibition is less severe than directly committing them. If, in addition to that, there is unlawful mixing of genders and abandonment of congregational prayers, the sin increases. Strive to earn a lawful (halal) livelihood, even if it means leaving work at the institution entirely, unless you are in a state of necessity where you cannot find any permissible work. In that case, you may remain until you find an alternative, choosing the least sinful option, as necessity is assessed according to its extent.