Is it permissible to deal with an usurious bank (such as Alinma Bank) to purchase a car, where the bank buys and owns the car then sells it to the customer at an increased price, with administrative fees? Is this transaction valid?

This is a permissible murabaha and tawarruq transaction, provided that the bank does not stipulate a penalty for late payment, there is no collusion for the commodity to return to the original seller, and the bank genuinely purchases the car, so it enters into its possession and responsibility before selling it to the one ordering the purchase.