Is it permissible to work in a usurious bank out of extreme necessity, knowing that refusing this job could lead to dire consequences such as the termination of residency and the absence of a source of income?

It is not permissible to work in interest-based (usurious) banks, except out of necessity. Necessity is determined by the occurrence of ruin, unbearable hardship, or the inability to secure a minimum standard of living. If the necessity ceases, the ruling reverts to prohibition. As long as you can transfer your sponsorship to a non-interest-based entity, there is no necessity to work in an interest-based bank, especially if doing so would prevent you from leaving it. Necessity is estimated according to its measure; do not exceed its limits. Therefore, fear Allah and seek permissible work, and supplicate frequently, for whoever abandons something for the sake of Allah, Allah will compensate him with something better.