You should actually be able to use the full Dropbox API from JavaScript. See https://github.com/dropbox/dropbox-js and https://www.dropbox.com/developers/datastore/docs/js.
> Thanks for writing in. It looks like the issue here is that our content servers (dl.dropbox.com) don't currently allow arbitrary cross domain access in JavaScript. (This doesn't apply to accessing the file directly in your browser, or downloading to your server locally, which is generally what one would do with the link returned by Chooser.) […] This is something that would need to be enabled on our side, so I'll be sure to pass this on as feedback.