Showing a project on an invoice is not standard practice, but I understand that it can be useful for your business processes. Businesses often attach supplementary documents to invoices when additional information is required to clarify matters for the client.
Anyway, I believe you can achieve this by creating a custom theme. You can import the current Plain theme and use AI to help you modify the theme code to produce the desired result. Once the custom theme is ready, set it as the default theme under Form Defaults.