Is working in a bank in a department unrelated to loans (delivering personal cards, credit cards, and remittances) unlawful (haram)? If so, is it permissible to remain in the job until another one is found due to necessity and the need to support the family?

If a bank is usurious, it is not permissible for you to work there in any position that involves direct engagement in the forbidden or assisting in it, such as helping to issue usurious credit cards. This is because a curse is mentioned in the noble Hadith for the one who consumes usury, its giver, its scribe, and its two witnesses. It is not permissible to remain in such work except out of dire necessity, where the Muslim cannot find other means to spend on himself and those he supports, along with the obligation to seek lawful employment.