To be completely clear, we do hide the mouse cursor as long as Resolume is the app in focus. The moment you switch to another app, there's not much we can do about the mouse. Because you just told your computer that we're not in control anymore.
If you're using Resolume normally, you will never see your mouse on the output.
Should you see your mouse on the output, you just need to click. Even when your mouse is on the secondary display, Resolume will regain focus, and hide the mouse.