Maximus Arcade runs in a window, although it doesn't appear like it because the Windows taskbar gets hidden and there is no window border. This is why you can move your mouse between monitors. However, MAME is running in full screen and the problem you're experiencing will happen when you run any software (which means primarily games) in full screen.
What you'd need to do is have MAME run in a maximized window which would let you control other software. However, this is not really a good solution, in my opinion, because your MAME window will have the typical Windows border around it, negating any illusion of authenticity on your cabinet, and doesn't look that great anyway. Also, if you're running at arcade resolutions (either with an ArcadeVGA or Soft15hz) then running in a window will not let you take advantage of that.
Basically, there's really no way to do what you're wanting without substantial tradeoffs on either side.