My Mustang PJ sounds thin with round wounds, especially with both pickups on. That is expected as they are now in parallel making a lowpass filter to ground, as with JJ pickups. (The body resonance is also higher than a standard P because it is lighter.) But the snap/bite more than makes up for it and so does the ease of playing. Please don't spend time and money modifying pickups or whatever until you try thicker sounding strings. I love the sound with D'Addario Chromes or nylon tape wounds. Both are available in short scale. Finger style or pick.
I liked the way it sounds with rounds, but Chromes certainly work well. I'm using Chromes now. You can still get a decent slap tone in the middle position if that's holding anyone back.
The Mustang for me is a portable and versatile bass. The chromes also help make it lower maintenance since you wont have to swap out strings often. It's a perfect bass for the studio when you put those on it.