I finally got this working by proceeding as follows:
1) Export the game as HTML5 to be embedded, without any modifications.
2) Create a separate page with an iFrame Element. Resize this iFrame Element (by script on rotation and resize events) to the window.innerWidth and window.innerHeight.
Now on the page with the iFrame, the game will occupy the full screen. This only works on the iPhone. It broke Fullscreen on Android, however. This is relatively easy to fix by adding the properties:
allowFullScreen="true" webkitallowfullscreen="true" mozallowfullscreen="true" oallowfullscreen="true" msallowfullscreen="true".
If anybody has a solution without the Full Screen API (which is broken on Firefox Android), let me know.