I’m on the hunt for a 5 string bass. I already own a 4 string Nash P bass, and I like it a lot. I have been wanting a 5 string and decided to take the plunge.
Going into a 5 string there are a LOT of things to consider just in that. Scale length, string spacing, fingerboard radius comes into play in addition to pickups and a particular sound. Fender isn't the only company out there making P/J style basses. And might there be other brands or styles that you might want to take a look at? Going from a 4 string to a 5 string takes some dedication for some (I think the majority of) folks. And it's amazing what you can grow into, being to like in 30 days if you were to compare two or more basses.
But let's get back to the P vs J thing. Both types can cover 80+% of what the other does in terms of sound. When you get into PJ's as you probably know, the placement of the pickups is as important as the type of pickups themselves. But then you get into pickup brands and their inherent sounds. If you go with a jazz bass will you need or want noiseless pickups?
P or J in a 5 string? Straight P basses with 5 strings is pretty rare. You're more likely to find a larger selection in PJ and a ton in J and even more that aren't P/J or PB&J
Another thing... Weight and balance. A 5 string with more strings can be more neck heavy than a 4 string.
One more thing... Your amp has EQ knobs as well as your bass. A GOOD active preamp can cover a lot of ground.
You haven't mentioned your budget.
Not all that long ago was I in your shoes looking for my first 5er. The only real advice I can give you is to make sure you play as many as you can and do not buy anything your can't play (hopefully with your own amp) that you can't return.
Having said that, rather than Fender I would point you to YAMAHA, if nothing else but due to out-of-the-box quality at great bang-for-your-buck price points. There is the TRBX series and the BB series. the BB series comes at 3 different price points. Sire is another brand worth looking at. I know they make a 5 string P for about $1000.
Good luck in your search.