I thought one way to potentially pass through 24fps would be to turn off upconversion altogether (using that same SHIFT RESOLUTION command). I tried this and I thought there was a way to display on the TV if I was getting 24fps or 60fps but I couldn't find it. Honestly, I'm not sure my eyes would catch the difference outside of a test pattern or something - particularly with the great upconversion of the the 5200.
Also, I think what you're saying about the GUI is mostly correct or correct for HDMI sources but take a look at the attached pictures. Menu is the full GUI menu overlayed over the main Wii menu and On Screen is just the on screen GUI with no menu. My TV overlayed the black bar on top indicating a 1080p signal. You can also see that the Wii menu does not take up the entire screen with a 1"-2" border around it. And this is all through a single HDMI cable (I also have an ethernet cable running the TV but I disconnected that just in case). I did change the upconversion to 1080i for the Wii though and then it filled the entire screen. And again, for some reason just cycling through the upconversion's output resolutions "fixes" the issue. Very strange.
Also, they do actually call it Bravia Sync in the manual despite not calling it that in the menu system. Not sure if it made a difference with my Wii issue but thanks for that tip as I definitely don't want that on.
I still think there must be a cleaner way to make this all work but at least I have everything working now without having to run a set of component cables through my wall.