This is a bit of an insane workaround, but couldn't you install an extra copy of AI War, then join your own game with that second install as an observer? Then, just ALT+ENTER if you're in fullscreen for each instance, and drag one window into the other monitor.
You wouldn't get a magical clicky map, but you'd at least have the galaxy map always viewable. And you may or may not be violating the spirit of the game license.