Recording payment made by customer to vendor

Hello Georgegaviria1,

Thank you for letting the community know your concern! 
As far as the open bill goes, since the vendor received the payment for that already, you can remove that open bill. Follow these steps to delete a bill:

1. Go to Expenses.
2. Choose the bill from the Expense Transaction Lists.
3. In the Accounts Payable register in the Charts of Accounts, find the bill and select it.
4. Within the report, choose the bill.
5. In the Bill window, select More at the bottom of the screen, and click Delete from the pop-up menu.
6. Click Yes, confirming to delete the bill.

For the customer's invoice, removing the total amount on the invoice should not affect your accounts receivable. However, I do advise getting with your accountant to confirm that this works best for your business. If you don't have an accountant, you could connect with a ProAdvisor specialist. They do offer a free consultation to businesses. Simply enter your zip code and search to find one near you. 

If you need anything else, let me know!

If we delete the bill... that would have an effect on our costs for the project. 

Not sure this is right  solution for us

Deleting the bill would have an effect on our costs. 

Changing the contract amount is not viable since we do need to issue final invoice for the full contract amount. 

There's no need to delete the bill, georgegaviria1.

All we have to do is mark the bill as billable. Then, we'll need to record the payment. Let me guide you on how to do it:

Before doing so, we'll have to turn on the Mark expense and items as billable option.

1. Click the Gear icon, and then select Account and Settings.
2. Go to the Expenses tab.
3. Tick Mark expense and items as billable.
4. Click Save, and then Done.

Then, open the bill and mark it as billable.

1. From the Expenses tab, select Vendors.
2. Select the vendor.
3. Go to the Transaction List tab.
4. Select the bill.
5. Mark the BILLABLE column.
6. Assigned a CUSTOMER.
7. Click Save and close.

Afterward, we're good to record the bill payment. We can do this by clicking the + New button, and then select Pay Bills. Select a customer. Next, select the billable expense. Then, click Save.

I'd also recommend reaching out to your Accountant so they can guide you with the process.

I've also added this article to know more how billable expense works in QuickBooks Online: Enter billable expenses.

I'm always around if you still need my help!