Without the use of a migration tool, you need to upgrade the entire site collection to SharePoint 2013, and then export your team site to a .cmp file. It is a fairly straight forward process even if it is a lot of work just to move a team site to 2013.
In your case I would follow the guide Upgrade a site collection to SharePoint 2013. The steps involved in this process are:
- Run the site collection health checks to verify the site is ready to upgrade. For more information, see Run site collection health checks in SharePoint 2013.
- Create an upgrade evaluation site to preview the differences between versions. (Optional)
- Upgrade the site collection.
- Verify that upgrade was successful and the site works as expected. For more information, see Review site collections upgraded to SharePoint 2013.
When all of this is done, you can easily continue with your task to export your team site in 2013 and import it to the real production site collection. It takes a while, but its done in a controlled manner.