Technically, yes, but a) that generally sounds weak, and b) it increases the difficulty of playing the chord because it means you have to either mute or pass over the first string.
Muting the first string is natural, trying not to mute it and fret it correctly is what makes it hard to play. It's vastly easier to just play the triad on the 2nd, 3rd, and 4th strings with the first 3 fingers; forget the bar. And if it sounds weak, turn up the volume or strum harder. :)
Perhaps you missed the smiley. In any case, you can drop the low root and add the high one and it still only takes 3 fingers. I generally leave out the high note and play the middle triad and the bass with the thumb Hendrix style as it feels the most comfortable.
But your main point was that the F was the most difficult to play, when what you really mean is your favorite 4 note version of the F is more difficult to play. A simple F triad, or even the triad with the bass is trivial to play, certainly no harder than any other first position chord except perhaps Em.
