Is it permissible to purchase a car in installments if the transaction involves the customer signing a loan agreement with a bank or installment company, where the loan amount is deposited into their account for a few minutes and then transferred to the car dealership, and the customer repays the installments to the lending entity?

This method is impermissible because it is a forbidden usurious loan. The bank pays the price of the car on your behalf, and then collects it from you in installments with usurious interest. Islamic banks do not follow this method. Your need for a car in your work does not make usury permissible for you, as usury is only permissible for the one in dire necessity, and there is no dire necessity in what you mentioned. There are many lawful ways for those who seek them, such as renting a car or buying a used car, and it does not have to be new.