First of all, let me say that I am absolutely sympathetic to your need to get outs in a game like the one you describe.....
That being said, I read your description of the play as a live ball, play on. From your description, the batted ball passed F3 and F4 (and I'm assuming F9) had no opportunity to make a play.
I've only hear the straight line between two fielders theory used in reference to base umpire positioning and when the BU is considered "behind" the infielders.