It gets worse, Chris. Even after you’ve hidden the edges, you’ve still got two big, rectangular faces that will cast big, rectangular shadows. Insofar as cast shadows are concerned, SU sees a transparent png background as opaque.
That’s why when people make a Face-Me or 2.5D (like the one you’re using) component from an image, they generally paint the image on a face that is cut to the approximate shape of the image and then has its edges hidden. The edges are hidden so the outline of the face doesn’t show and the face is shaped like the image so it will cast a reasonably shaped shadow.
-Gully