Unfortunately, styled-based variables won't work across different documents. Nor will work trying to get Section Marker, that are also document based...
So, AFAIK, you must find a workaround in order to achieve this missing feature.
1- Create a custom text variable. Enter manually the name of the section. Insert this variable in the docs Master pages where you want the name of section to continue. Apart for time consuming process of manually entering section names, an other obvious problem is that you will have to create a variable for each section. Note that you must make sure that "variables" are checked in the book Sync Options.
2- The one I would go for (it's a dirty cheat but...)
Based on the example you provided: in your Doc 2 Master page, go Text > Insert special character > Marks > Section Marker (assuming you have defined them for each section).
Doc 3 to 7: place Doc 2 as a graphic in your Master pages. So, if you change the section marker in doc 2 and save, all you have to do is update links. See example below: