Full refund with an apology for having not accurately quoted to begin with. The customer does not owe you anything (certainly not cost of check and postage) since you gave them a price and you are the one not able to meet it.
Just like buying anything (car, appliance, clothing). Picture yourself as the customer in that situation. You are quoted a price. You agree to the price. Then after the transaction has been agreed to, you are told "oh, gee, sorry. It's more than I told you." How would you feel? Not to sound harsh, but some might feel that was "bait and switch". I know I would.
If you fully refund with apologetic explanation the customer is less likely to come away with a negative impression. After all, he did nothing wrong. If you charge him for anything, he will likely tell others not to do business with you.