So I applied the flat fee (one time) but the work entries show a billable amount on the invoice. How do I fix that?

You might have had the project set to Hourly at first so when changing it to Flat Fee it will not revert all past entries to non-billable. Follow the steps below to make your entries non-billable

1. Click on Invoicing from your blue menu bar
2. Choose the desired invoic.
3. Make sure that the invoice is not yet finalized (If you see the Unfinalize button, click on it)
4. Click on a work entry that is showing billable
5. Uncheck the box next to Billable
6. Click the Save & Close button

Repeat Steps 4-6 for each lined entry except for the one that applies the flat fee. By the time you're done, the invoice balance should just be the flat fee.