Is it permissible to borrow from usurious banks to pay for family residency renewal fees, given the inability to pay and others' refusal to lend? And what is the legitimate alternative?

Dealing with usury is one of the gravest sins, and Allah has declared war on those who engage in it. It is destructive for its perpetrator, and accursed are its doer, its scribe, and its two witnesses. A believer must avoid dealing with usury and even contemplating it, for Allah grants those who fear Him a way out, relief, and sustenance from where they do not expect. The questioner should present his problem to the competent authorities and charitable people, and he will find a satisfactory solution if he fears Allah and strives to solve it through permissible means.