Is it permissible for the lender to reclaim his money from his debtor brother, knowing that his brother will borrow from a usurious bank to repay the debt, and does the sin fall on the borrower alone or on both of them?

You have done well by lending your brother a benevolent loan (Qard Hasan (definition)). If he insists on taking an interest-based loan (riba) and pays off your debt with it, there is no sin upon you for taking it, and the sin is upon him, because its prohibition is tied to his responsibility. Allah has forbidden riba and has warned those who practice it with war from Him, and He has cursed its consumer, its provider, its scribe, and its two witnesses. It is not permissible to engage in riba except in cases of recognized legitimate necessity, which is when a person fears destruction or extreme hardship, and only to the extent that such necessity is repelled. There is no necessity in what you mentioned that permits your brother to engage in riba, so he must fear Allah.