Is it permissible to borrow a sum of money from someone who himself borrows from an interest-based (usurious) bank, knowing that I will return the amount to him without any increase (non-usurious)?

It is permissible to take a benevolent loan from a person who deals in usury, and it is permissible to conduct any permissible transaction with them. Sheikh Ibn Uthaymeen mentioned that dealing with someone who deals in usury is permissible as long as the transaction itself is sound and permissible, citing the Prophet's (peace be upon him) dealings with the Jews who consumed ill-gotten gains. He accepted their gifts, responded to their invitations, and bought and sold from them. In summary: There is no harm in conducting a permissible transaction with someone who earns unlawfully.