You are getting the "responsive design" the wrong way. "Responsive design" is unlike a different design/version of your site for mobiles/tablets. It's the same site with one design working for all platforms. A "full site" button I think is not possible since there are no two versions, just one being viewed differently.
To get a better idea of what a responsive design is, read this -
http://johnpolacek.github.io/scrolldeck.js/decks/responsive/