Nor is it about hanging Christmas lights, which my wife just got done making me do.

I can't see a way around it... it looks to me like the code that renders in the non-shadow part is different from the code that subtracts the background pixels from the pixels representing objects. So, the rendering program paints both background pixels and object pixels white, but it's still only the background pixels that become transparent on export.

I think the only option you would have would be to import the image in to PSP, and remove the all white pixels with the selection tool. 

Sorry I couldn't be more help.