the banks wont be able to give you an auto loan for a car you already own. they will have to give you a personal loan at a higher interest rate.
you need to get an auto loan first. they give you a huge check for the exact amount, you pay the seller, get the car, and then you can pay back the loan on your own terms.
just call the bank and figure out the process so you and the seller can have paperwork ready for it.
if you do it the other way it will be much harder with less benefit and flexibility.