You'll want to create your customer as a supplier and add a special character to identify it as a supplier. Let me guide you how.
Go to Expenses, then Suppliers.
Select New supplier. Then, enter the supplier information.
Make sure to add an identifier in the supplier's name so QuickBooks won't prompt it as duplicate.
Then, create a journal entry to record both payments on your A/R and A/P accounts. This process will let you apply the payment to the existing bill and invoice, so you can just either "receive payment" or "pay bills" on the remaining balance.