Is buying a car through the bank—by paying a down payment to the showroom, then the bank settling the remaining amount with the showroom, with a sales prohibition on the car, and paying installments to the bank with interest—considered unlawful or doubtful?

If the bank buys the car and takes possession of it, then sells it to you at a mark-up (Murabaha) with installments, and does not stipulate an increase in case of late payment, then the transaction is permissible. There is no harm in the bank preventing the sale of the car before the installments are paid off.