Is it permissible to purchase a car by paying a down payment to the dealership, then the bank pays the remaining amount with an added profit margin, and I, in turn, repay the bank in installments?

The aforementioned transaction is not permissible, because the bank lends you the remaining car price, with the condition that you repay it in installments with an increase. This is precisely the forbidden Riba (usury). The solution is to search for a bank that conducts Islamic Murabaha (cost-plus financing) in accordance with the Shariah-compliant regulations outlined in Fatwa no. 31758 and Fatwa no. 56213.