I thought they gave it the F designation because of coolness factor, nothing more...
I vaguely remember this on the discovery channel....
..just googled this:
"The F-117 program was designed to employ the best fighter pilots, and the explanation given (on a recently televised documentary) by a senior member of the F-117 development team was that fighter pilots would be reluctant to fly a "B-" or even "A-" designated plane. There has been something of a class system separating fighter and bomber crews, particularly in the days of the Strategic Air Command (disbanded in 1991), and flying one type often limited a pilot's future prospects for flying the other."
At this site