I am not near a „real“ computer…so I can only give advise. So it is not possible to put the fullscreen button into hype in only one of the layouts and delete it from the other? That would be easy.
If the button is not part of your of hype you need to know a bit of coding as you need to find its ID or Class. Google „using the webinspector“ in that case and at least you can learn how to find the ID of an object on the page on the weekend.