How can I use a fullscreen app (specifically a Keynote presentation) while displaying another app (specifically a movie) simultaneously?
Unfortunately, you can't.
Apple's splitscreen simply doesn't currently support that. Unless the app specifically supports it's fullscreen mode in splitscreen, only the windowed mode will work (if it's supported there.)
With that said, you can export a Keynote presentation to another format like QuickTime which supports splitscreen playback. You can then run this in a window with a splitscreen, and the relevant other app (such as the movie) on the other side.