It is fantastic that you have been so fortunate because you listened to people that had the right idea.
But that's the thing: THEY had the right idea.
The OP's band director is indeed the one in charge, but that doesn't mean he shouldn't be spoken with. Jazz is about the music, not the looks. The music is better when the players use efficiency to their advantage, combined with technique of course. If this means switching to a six string, then he shouldn't be limited to that. If it makes his playing better, then it's better for the music.
The music is what counts. Yes, image is a factor that shouldn't be ignored, but that doesn't mean it should interfere with efficiency in playing.
Yes, there are the six string players that tend to stay in one position. But Spade stated that he is clearly able to slide around plenty, so that's not a problem.
What I'm basically saying is this: The band director wanted the OP to go to a four string because
And that, because of the fact that it has to do with image instead of the music, is not a valid reason to make him move from a six back to a four.