In my opinion, they are just being jerks.
They have not spent any money on your file that you need to pay them back for. Up front application fees are a ploy to make you feel roped into the transaction and less likely to shop around.
You should name the lender.
Tell them to produce an invoice detailing the charges that you owe. They won't produce it because they can't.
Good for you by calling the credit card company.