I was a bit confused when the shadow catcher didn't work with ezdome. After fussing with it for awhile, I realized that it doesn't work the same way in Superfly "Cycles" as it does with Firefly at all. Instead of rendering as a transparent shadow on the prop surface, it's rendering the shadow over the "background" shader--whatever that happens to be.

While that works when using "background" as the sole background of the rendered image, that has a few issues of its own. The biggest drawback is that the "environment texture" node gets displayed as a regular image in the preview, so you get no sense of camera orientation relative to the background without rendering first. Using a background prop, such as the one with ezdome, doesn't have that problem. The preview will properly show the same orientation and perspective that you'll get in render, no random guessing required. 

That of course leaves me back were I started... I tried to work out if the two can be Frankensteined together, but the orientation between them was completely off. I came across a tutorial, of sorts, in the Poser manual that gave some pointers. It's in the section Other Root Node Types, under "BG Environment". Using a positive value for the rotation offset (0.000000, 1.570000, 0.000000) in the Mapping node does the trick. The only noticeable drawback is that the "shadow catcher" prop leaves a faint black outline.