Can Flute of the Outer Gods be used for monsters "not on the board" like in The Terrible Experiment?
No,

While I'd be tempted to allow this as it would be a very cool thing to do and it is hard to find a good use for the Flute of the Outer Gods, this question is addressed directly in the FAQ, pg 10.


  Q: Do cards that affect monsters at Miskatonic University
  (such as the “Campus Security Increased” Mythos card or
  the “Flute of the Outer Gods” Unique Item) affect monsters
  spawned by “The Terrible Experiment”?
  
  A: No. Those monsters are not actually on the board; they
  can only be fought as if they were.