Unfortunately, the answer is 'no'. I've been trying for some time to work this issue out, and the only way I've managed to make it work is to have the main window (the first window that is displayed when you start the Tournament Director) display the Tournament screens, and when you want to access the Settings, open it in a modal dialog (modal being the kind of dialog in which you cannot access the window "underneath" (the window that opened the dialog) until you close the dialog).
So, essentially in the next version you should be able to have the Tournament window on one screen (displayed to players), and the Settings dialog open on the other screen. You can open and close the Settings dialog at will, or simply leave it open the entire time. However, you cannot access the Tournament screen while the Settings window is open. Therefore, right-clicking on the Tournament screen to use the context menu, or using hotkeys associated with the Tournament screen (like X to bust a player out, etc), won't work while the Settings dialog is open. You'll have to close it first. In order to deal with this, I've added a "Control" tab to the Settings dialog, which simply contains all of the controls (in the form of buttons) you should need to run your tournament, like busting players out, pausing the clock, rebuying, etc, etc. So, theoretically, in this example you should be able to run your entire tournament from the Settings dialog on your laptop (or desktop, whatever), while the Tournament screen is always displayed on the extended screen (to your players).
I have already tried this and it works well. I understand the issue with where dialogs open up. And now that you've mentioned that, I just realized that the preference to "save" dialog locations works fine, whether you close a dialog on the main screen or on the extended screen (it will remember correctly - at least it did on my laptop - different laptop manufacturers could implement the "extended" screen differently), so I need to look into seeing if I can better control where they open. Would be nice if I can get all dialogs to simply open up on the screen on which the Settings dialog is located...