Hello Phil,
Sorry for the delay in getting back to you. We do not currently offer a duplication feature for Cloud Platform projects. You would have to create a new project and deploy your application into this newly created project. You can do this via gcloud by following the instructions here.
The process of duplicating your current database depends on the import/export instructions offered by Google for the type of database you are using in your project. Since you are using Objectify I will assume it is Datastore. Datastore does allow importing and exporting of entities. You can find the guide how to perform this task at this following link.
Also, if you have any specific roles or service keys associated with your deployed project then you will have to duplicate that for your new project as well. For instructions on managing service accounts and how to authenticate roles for users, you can follow the Google documentation for it here.
If you are unsure about your current service accounts and keys information related to your deployed projects, you can navigate to “IAM & admin” portion of your Cloud Console. Google offers a set of roles that project owners can assign for their App-Engine applications.
Since you said you updated the Google SDK already then it would be recommended to update to Objectify 6 (as this version uses the latest Google-provided SDK to access Datastore). The key differences between Objectify 5 & 6 are listed on their GitHub link as well as how to upgrade between the versions.