Hello,
Based on the issue detailed, in particular that other apps also fail with similar errors, my guess is that PWA does not have access to the App Management Service itself. So in other words rather than it being a permission problem in the PWA configuration, actually the permissions configured in the SharePoint farm for Project Server and apps are not correct.
Check the permissions granted to the Project Server service account configured in CA and compare it between DEV and UAT, specifically what permissions are granted to the App Management Service and database(s) configured.
From memory the Project Server service account will need explicit access to the app management service.
Hope that helps you find the cause of this issue.
Regards,
Martin