No, there is no call in LMS that is the exact equivalent of CompleteSale, however the LMS SetShipmentTrackingInfo call is a subset of the CompleteSale call. CompleteSale enables a seller to do various tasks after an item transaction or multiple-item order has been created. A seller can leave feedback for the buyer, change the paid status, or set shipment tracking information (or any combination of these).
SetShipmentTrackingInfo, as the name suggests, allows the seller to set the shipment tracking information, which is also allowed by the CompleteSale call.
CompleteSale can be processed if either ItemID and TransactionID or OrderID are specified. SetShipmentTrackingInfo requires the LMS OrderID and OrderLineItemID as the input.
The concept of an 'Order' is only used in the Trading API when a buyer makes combined payments. Each payment is a transaction. But in the Large Merchant Services solution, every sale is considered to be an Order. It is important to recognize these conceptual differences when you are working with multiple APIs.
OrderLineItemID is a based upon the combination of the eBay Trading API's ItemID and TransactionID fields. The number before the hyphen is the ItemID and the number after the hyphen is the TransactionID.