Mercury's response: The POS program needs to be able to send a Void or Return request for that transaction.
A little more research into the Partial Approval process shows Mastercard and I believe Discover both require this type of approval. In the event the customer cannot pay the balance, a Real Time Reversal is performed which is basically a void sent from the merchant.
So, what is the best way to close out a sale that is partially paid for?