I believe that Photoshop can transform an image from the usual rectangular projection, to spherical, and you may be able to apply that to the SkyDome. If you look at some of the other pages on that website, you might get some other ideas on how to use it.
A SkyBox might be easier to work with for this purpose, but you might still need to pre-distort the 6 images for the box so the result would look undistorted.
Something worth looking into might be Panocube. It expects a spherical panorama as an input, but it comes with some extra tools that might be of some use. http://www.panoshow.com/panocubeplus.htm