Hi Brian,
The only way to verify that a line is paid it by verifying that the invoice has been paid. You can do this by creating a linked field on your InvoiceLine table that links to the BalanceRemaining field on the Invoice table.
You’ll need to create a button on one of your screens with the Generate Report action for generating your report. You can setup the filters on the action itself for only returning lines specified by the filter. You can filter for the particular item name and your linked Invoice.BalanceRemaining field equals 0 like so.
Give this a shot and post back if you have any more questions. I hope this helps.
-Audisho