The better instrument is the one that makes your job easier and makes you play better. If you think the six is a better sounding bass and it makes positions easier for you, then use it. While its probably uncommon for a high school jazz band to have a 6 stringer, its not at all a bad thing. Why don't you explain that they are the same instrument, one with extended range. Tell him you like the position possibilities on the 6er and the options it opens for soloing. I wouldn't quit, even if they make you play a 4, music is filled with idiots, get used to it now. Good luck.