Not sure how much of a grant you're talking about You can use any bass, though the JB is a good choice (a P can get those nice mids as well). My son played in a jazz band with my 1999 P Lyte Precision (which has both Jazz and P pickups) and it was great. You might want to consider a JB just for the sleeker, faster neck, but then again that depends on your playing style. BTW, if you're looking to go fretless, those standard Fender fretless JBs are great tone-wise (even rivaling the more pricey Jaco).
Lone Wolf Club #2, Fretless Club #714, Flatwound Club #300, Danelectro Owners Club Member #37