Is it permissible to reserve a car with a small deposit before the bank approves the Murabahah request, and is there any prohibition in doing so if the approval is not granted?

It is not permissible for you to reserve the car by paying an amount to the owning entity before the bank approves the Murabaha transaction, because there is no contractual relationship between you and the owning entity; the relationship will be between the owning entity and the bank. Any previous contractual link between the client commissioning the purchase and the original seller must be cancelled. So, if you have reserved the car, any previous agreement must be cancelled and the paid amount retrieved before the bank carries out the Murabaha operation.