I wouldn't worry: I have the same problem with Jazz basses. LOTS of players - be it people I know, or well-known bass-heroes - have played great basslines on them. But I can't seem to make the sound work with my playing style. Give me a Precision or a Thunderbird and I'll be happy as Larry, but for reasons I can't fathom, I've never been satisfied with the tones I get out of a Jazz. It's just something I've had to accept!
That said, have you ever tried a "super-strat"? I've seen H/S/S setups, and twin HB setups - the former might do something for that "shrill" bridge pickup, and you still get the look and feel of a Strat.