The following code will set the background-image style of the document within the editor (where ed is a reference to the current TinyMCE instance)
var edObj = ed.dom.getRoot();
//Set the Style "background-image" to the button's color value.
ed.dom.setStyle(edObj, 'background-image', "url(some_background_image.jpg)");
Note, I've only tested this on Safari and Firefox on Mac.
The only problem is, even if you are editing the entire HTML document using the fullpage plugin it still won't return the document with the style applied when you retrieve the content using getContent().
If you load TinyMCE with a document that has the background-image style already defined, then TinyMCE will render it.
With this in mind, depending on how you provide the end-user the ability to select the image, one approach would be to provide a way to send the selected image details back to your server at the same time as you save the document and process it there, applying the background-image style.