Assuming the Account is related to the Project, what you could do is add a calculated field in Studio and populate it with the billing state from the Account:
Go to your Project Module in Studio, add a plain text field, click the "Calculated Value", and "Edit Formula"
On the top of the small formula window you will see an option for "Related Field" click that
A small "Formula Builder" window will open, select Account, and Billing State.
Select Insert, Save your formula, and your field.
Add your newly created field to the layouts you want to see it in: Record/ List.
The field is a calculated field so it will populate when the Account is selected in the Project record.
Should you want to update it for existing records you can go into your Project Module's list view and select the records you want to update (as if to mass update them), then choose "Recalculate Values".
This will update your newly created field (and any other calculated fields) for the selected records.
Hope this helps,