For the semantic model design required for the customer satisfaction report, the choices for data source authentication method and mode should be made based on security and performance considerations as per the case study provided.

Authentication method: The data should be accessed securely, and given that row-level security (RLS) is required for users executing T-SQL queries, you should use an authentication method that supports RLS. Service principal authentication is suitable for automated and secure access to the data, especially when the access needs to be controlled programmatically and is not tied to a specific user's credentials.

Mode: The report needs to show data as soon as it is updated in the data store, and it should only contain data from the current and previous year. DirectQuery mode allows for real-time reporting without importing data into the model, thus meeting the need for up-to-date data. It also allows for RLS to be implemented and enforced at the data source level, providing the necessary security measures.

Based on these considerations, the selections should be:

Authentication method: Service principal authentication

Mode: DirectQuery