I think the sound engineer has more to do with how the bass sit in a mix more so than the bass itself. I’ve played PBs for almost 40 years or better. I also have a Warwick (active, but alway set to flat) and a Jazz Bass. I can honestly say, you can get as deep as you can stand with either. You can also eliminate the roundwound growl with EQ as well.
Personally, I like recording with fender basses with volume and tone control wide open and direct to the console with no effects. This give the engineer a full tonal palette to work with. The Warwick set to flat.
Just a tip. If a bass has it’s low mid scooped to make the low frequencies more noticeable, the more you scoop, the more difficult it will be to hear in the mix. And I can assure you a Jazz can have as much fluff/fuzz as a PB if that’s you want.
I’ll agree there is a difference between the voice of a PB and a Jazz, but depending on the EQ, there really is a lot more commonality than difference in the hands of a skilled engineer.