I'd say the clue is that the P bass "punches through the mix brilliantly" using your rig and then the J doesn't.
If you get a problem always start by looking at what just changed. In this case it's the Jazz.
What Jazz is it, what pickup does it have, is there a preamp, what strings are you using and how do you EQ it.
Then again, what is the rehearsal space like? Small box room with everyone playing at "big stage" volumes? Does the jazz get lost on a gig?
I use an Aguilar TH500 through 2 Super Compacts, which is the modular version of your cab. I don't even change the EQ when I switch from a P bass to a Jazz and they both sound huge.