Is it permissible to purchase a mobile device using my brother's credit card and pay him back monthly, knowing that the store owner offered the option of purchasing either in installments or with cash, and that dealing with usurious banks is prohibited?

You have done well in abandoning dealings with usurious banks. You are permitted to purchase the device using your brother's credit card if it is Islamically permissible and you both adhere to the Shariah-compliant regulations for installment sales. Otherwise, it is not permissible, as it involves acknowledging and cooperating in wrongdoing. You must advise your brother regarding the impermissibility of this card.