Welcome to the club Max!
Adjectives like thump and growl can get very subjective, but that's besides the point. I think it can be done, but you have to decide on an construction method (set neck vs. neck through) and an electronics package that will dial in the right amount of each characteristic.
As far as construction method, things to consider are the woods on both the Fender & Warwick. Does the P bass have a maple or rosewood fretboard? What's the main body wood on the Warwick? Do you want increased sustain & mid-highs (neck through)? IMHO, A slightly scooped midrange (set neck) will help get you close to what you're trying to achieve, but you might be better off going with a neck through (for some of those P-bass characteristics) and letting the electronics dial out what you're not using.
Off the rack you might find the thump of an Alembic to have more clarity and resonant focus then a split coil P-bass pup. When it comes to dialing both sounds you might want to consider the Signature or Anniversary electronics which give you a volume & filter for each pup....
Hopefully that's enough to chew on for now~ just keep in mind that anything is possible with Alembic. Just dig around on this website, talk to the club members. Hopefully you'll get a good idea as to what's going to suit your needs....
-Mike