Is it considered usury if a car dealership grants an initial down payment to a buyer, and the buyer then uses it as a down payment to the bank that buys the car and pays its price to the dealership, with the buyer subsequently repaying the car's price to the bank?

What the questioner has mentioned is not sufficient to categorize the transaction, and the nature of the amount given by the showroom to the questioner as a first installment is unknown. If it is a loan that obligates the questioner to purchase from the showroom, then it is a loan that yields a benefit to the lender, which is impermissible. If the showroom desires the price of the car immediately, and the questioner does not possess the price and wishes to purchase it through the bank, there is no permissible way for this except through a Murabaha sale for one who commands a purchase, whereby the bank first buys the car, and once it owns it, sells it to the questioner in installments.