There is no direct link between an item fulfillment and an invoice through the API's. The Order Exchange determine which item fulfillment matches which invoice but looking for the item fulfillment that has all the item and quantities that match the invoice. This is also done based on ordering them by the shipment and transaction date as 2 invoices may have the same item and quantity on them.

It is therefore essential that you invoice each fulfillment in full, and one fulfillment to one invoice.