As of now, BQE CORE APIs do not provide endpoints for creating objects via templates, e.g., there is no endpoint to create projects from templates. You can only create projects from templates through the web app. However, as a workaround, you can set up pre-built JSON objects as reusable payloads that could act as project templates and then send them in POST requests while dynamically inserting some of the values like name, clientId, contractType, etc.
To set up phased project templates, you could again create a set of pre-built JSON objects with the required structure. Just make sure to create the root project first, so that you can then set its identifier as the rootProjectId when creating its child phases. Check out the API Documentation to learn more about the structure of the Project object.