Hi, Menno!
You can set a default month for the reporting environment. This default month actually is effective for all three environments (dev, staging, and prod), but as you've already pointed out, dev reporting months are preserved via session cookies. In production, the user will always be taken to the default month first, regardless of where they ended their last session.
While in the dev environment, click the Project tab at the top of TBM Studio. There is a button called Time Settings.

In the window that appears, check the Enable Default Time Period checkbox, and then choose a default time period.

These will not take effect until you check in your changes. After you do, wait for staging to finish calculating, and then promote to production (if you want these changes to take effect right away). When you promote, whichever month you have in the Default Time Period field will be the first time period that users will see in production when they load Apptio.