Viable option, maybe??
Vern, It is hard to tell from you pictures of the rectangular duct if it is at least 4 inches wide or if you were able to actually remove the bottom portion of it to clean it and photograph it. Would it be possible for you to get some 4 inch rigid pipe and assemble it section by section until you could slip it into the 4 inch hole up at the top? That way you would then be directly connected to the roof cap and the full force of the dryer fan can exhaust right up to the cap? You would need to put the crimped end up into the duct and add additional sections until it is able to be transitioned on to the dryer exhaust with a 90 degree elbow at the bottom. You may have to slit the top section of the crimp so it slides into the top piece more easily.
Think it's do-able?