Hi Saratgu,
You’re right, entering the billing info will move the balance from the Unbilled Purchases to Accounts Payable on the next financial exchange. This movement is recorded via journal entry in QuickBooks Financial.
I would recommend reaching out to QuickBooks Point of Sale Support to check your setup and review transactions. They can use a screen-sharing tool as they find out what happened to your transactions.
If you have other questions in mind, you can always give us a comment below.