Hi Thomas,
I've just finished working on some viewers with a normal 360 viewer with the option to toggle the split view to use with Google Cardboard.
The way I did it was to create two divs, 1 has the regular view and the other has the split view. I have UI menu for cardboard which hides one div and unhides the other. It also loads the necessary split view data for that specific view.
Ill try and break it down.
I have a 3D array which stores the data
[regular 360 view - 1][split 360 view - 1 - left][split 360 view - 1 - right]
[regular 360 view - 2][split 360 view - 2 - left][split 360 view - 2 - right]
...
...
All views, 360, split left and split right all use the 1 set of images.
When the user switches to the cardboard view (split view) it loads the left and right views from the array depending on which regular view you are look at. It also fullscreens the div and enables mobile device gyroscope which I merged from the Device Orientation demo.
My implementation is just a set of views with no tours so I'm not sure now well this will work for your needs.
I hope this gives you an insight to a possible solution :)