Timing of asking for the fill should have ZERO effect on the smoothness of transition.

You don't have this problem simply changing variations (or at least, shouldn't), no matter where in the bar. The arranger, should you ask for something fractionally late, should give it you on the next beat or sub-beat (8th note) in perfect time. The only exception to this is when you change chords during a longer note, and the arranger seamlessly bends the note to where it should be instantaneously (in case you are fractionally late with the chord change). But rhythmical accuracy should NEVER suffer.