A surcharge is added to the total cost at the time payment is made. A surcharge invoice is generated at the time of processing  in Xero or QBO.

Does Wise-Pay handle it?

Wise-Pay automatically does handle Payment Surcharges. One of the core capabilities is for Wise-Pay to automatically handle Surcharges, as well as managing payment deposits directly in your accounting package. Your customer is able to download a copy of the Surcharge Invoice from the Actions in Past Payments widget.