The code that you linked to (from spazzarama), which you said you were using in your project, captures the front buffer of your DirectX device. Have you tried capturing the back buffer instead? Going from the code on your linked site, you would change line 90 from
device.GetFrontBufferData(0, surface);
to
Surface backbuffer = device.GetBackBuffer(0, 0, BackBufferType.Mono);
SurfaceLoader.Save("Screenshot.bmp", ImageFileFormat.Bmp, backbuffer);
This would also involve removing lines 96-98 in your linked example. The backbuffer might be generated without the obstructing window.
EDIT
Nevermind all of that. I just realized that your linked sample code is using the window handle to define a region of the screen, and not actually doing anything with the DirectX window. Your sample code won't work around the obstruction because your region is already drawn with the other window in front of it by the time you access it.
Your best bet to salvage the application is probably to bring the DirectX window to the top of the screen before running the code to capture the image. You can use the Wind32API
BringWindowToTop
function to do that (http://msdn.microsoft.com/en-us/library/ms632673%28VS.85%29.aspx).