I think I need to realise that a Jazz isn't a P Bass, so stop trying to use it as one and don't be disappointed when it doesn't sound like one too.
I've restrung it with DR Sunbeams and the round wounds make all the difference for me and it fills that gap sonically that I didn't have with my other basses.
So I now have a P Bass with flats, a P Bass with rounds, a Fretless and a Jazz. I think that should do me for a while at least.