The Ultra J are quite similar to Sadowsky pickups. Very bright and punchy. But as you say, you can get something of a bright P-bass tone with soloed and tone rolled off. I bet in series it would be a punchy deep monster.I haven't tried the Model J so not sure how they compare, but the Dimarzio Ultra J pickups can be similar to a P with the neck solo and tone rolled off.
I really think you should try a series parallel switch with the stock pickups first. A push-pull pot is like 15 bucks and there's a pictorial wiring diagram on Semour Duncan's website. It will give you plenty of punch and if you want some more snap you could get some stainless steel strings.I've got a Tagima Jazz bass that plays great, but needs some more sonic punch. The current pickups are generic "vintage" J's and lack punch.
I generally play with flats, and I play a Mustang with my band, but I practice on the JB and would like to gig with it too, as it's a very smooth player.
The main thing I'm looking for is a more powerful, punchier neck pickup with a slightly P-ish sound. I know that the Model J is the default for that kind of thing, but I may want something brighter to go with flats and a bass that's not the brightest.
I also want to keep it under $150-200 for the set, because that's what the bass itself is worth.
I've put Series-parallel switches on at least 4 jazz basses and never regretted it. Two cheap pickups in series often sound better than the sum of their parts.
All that said, I played alot of country music with a Jazz bass with Model J's and a series parallel switch.