I left a comment earlier in this thread mentioning that you may want to try rounds instead of flats, but wanted to give a quick follow-up/caveat to that:
While P w/ rounds and P w/ flats are certainly 2 different flavors, and using one or the other may make a difference to you.... neither are going to emulate a Jazz Bass very well. A P with rounds will have some more of that grindy/zingy character that you might also find on a jazz with rounds, but a J is still going to be your best bet to actually sound like a J.
I personally find it much easier to emulate a rough P sound through a J neck pickup than to emulate a J sound through a P pickup. Though, the J neck pickup method still doesn't quite check all the boxes I associate with classic P-Bass goodness. It'll do a respectable enough job though, if you don't have a P available.