Is the transaction conducted by a bank, whereby it purchases a commodity specified by the client and then sells it to him at a higher deferred price to be paid in monthly installments, permissible? And if it is permissible, and the price of the commodity is greater than the loan offered by the bank, is it permissible for the client to pay the difference to the store to complete the purchase?

If you inform the bank that you will buy a part of the car, there is no objection to that. However, deceiving it and giving it the impression that you will buy it in full is not permissible, because it involves lying and cheating. This is also because the bank buys it in full and then sells it to you in full. If you owned a part of it, it would be as if you are selling this part for cash to the bank and then buying it back from it on credit for more, which is usury (riba) and the forbidden Bay' al-'Inah (a type of buy-back sale), in addition to the apparent fictitiousness of the transaction. You can ask the bank to buy the car for you and pay an advance installment, or tell it that you have paid a part of the car's price to its owner.