I think (and I may be wrong here) that even though you see an image when you post to Facebook, it's still technically right clickable from FB, but it's a smaller, compressed image. Facebook is just creating a link with a version of the image you have here.
I suppose Twitter can't see or won't generate an image to display when you direct link, which is interesting.... wonder if it has to do with the 'no right click'. Does it work from your FAA profile?
I use a service called Buffer to manually create and schedule tweets, and I just upload screen res (even smaller than the 900px images here) with links to my artist wanted. It's an extra step, but I put hashtags and text in my tweets anyway, so I don't find it to be a dealbreaker.