Extra-solar identification of binary planets Pluto and Charon orbit about a common barycenter that isn't inside of either planet. Let's assume that this relationship was scaled up to the point where it could be detected by a search for extrasolar planets. Could any of the methods we currently use tell us the difference between one massive body vs two bodies of nearly identical mass?
Yes indeed there could, especially in the case of a transiting planet system. Whilst the radial velocity curve would be nice and smooth, reflecting the position of the barycentre of the double planet, the transit light curve would show structure associated with the two planets obscuring the disc of the star. Essentially the transit shape would vary depending on the orbital phase of the double planet and on occasion the transit would occur in two steps as first one planet then the other 
passed in front of the stellar disc. No such objects have been found.

People are already looking for this in the context of searching for "exo-moons" - for example,
the "Hunt for Exomoons with Kepler" programme. Finding a double planet with equal sized components would be comparatively easy compared with the tenuous signals expected from exomoons.