Can Beat Cop’s free triggered ability be used during any player window to incur 1 damage to an enemy at its location, even a player window occurring during another investigator’s turn?
Yes. Any player can use free actions, like the Beat Cop's ability, during any player window, including during another player's turn.
From the rules reference.
Free triggered abilities ([free action symbol (squiggle)] triggered ability may be triggered as a player ability during any player window.
The red boxes are player windows. Players may use [free action symbol] triggered abilities in these windows.
See pages 23 and 26 of the rules reference for the timing of the player windows.