I agree there's no such thing as a "pick" bass, but I think good semi-hollow basses have an especially righteous, cool tone with a pick, such as the Epiphone Jack Casady someone mentioned. That's a good call. I had a DiPinto Belvedere for a few years, and it was great with a pick: huge, but with some definition. Check out the live QOTSA DVD to hear their players tearing it up with picks on old hollow or semi-hollow Yamahas. I got a similar tone on the DiPinto.
After many years of fingerstyle, I play mostly pickstyle now due to a joint problem in my right index finger. I love P-basses, but for some reason usually prefer J-basses with a pick. Maybe it's because they don't "clank" quite the same way, and have a certain "bite" I don't get from a P-bass. Or maybe it's because with the treble turned down a bit, they still have lots of definition. Not sure exactly, but it works for me.
Regardless, good pick tone really is mostly about touch and technique, and the pick itself makes a big difference. Try several. You really might be able to get the tone you want out of your P-bass. (Not saying you shouldn't get another bass, mind you. We all know you probably should, and we don't need an excuse like allegedly requiring a "pick bass." Just say you want another. It's okay. We understand.

)