I have a fresh idea. It is possible using AppleScript to tell the Dock app to do things. The dock app has a way to move apps between Spaces. I have a hunch that it should be possible to take advantage of that to switch an app between spaces. You can see some AppleScript code to manipulate the Dock on this page.
I rarely use AppleScript, and I've never used Spaces, and I've never even thought of using the Dock for any reason with KM. However I have a good feeling that this may contain a viable way to solve your problem. You should do some research in this direction, and probably someone on these forums can help.