I own an American Professional Precision. Antique Olive, rosewood. I've been using it for about 5 months now, almost exclusively for recording + a couple of rehearsals. I have not used it live yet (and probably won't).
I really liked the V-mod pickup a LOT at first. The lows aren't overblown like on certain other P-pickups. The lower mids have just the right amount of punch. The upper mids and highs are very sweet. But maybe just a little TOO sweet. The V-mod lacks that little extra bite to cut through in a dense mix in my experience. It has a tendendy to get lost. It sounds really great soloed though. Very detailed even when the tone control is pulled back for a FAT sound. But then it kinda gets lost in a dense mix without extensive eq'ing. And a P-bass that doesn't sit nicely in the mix is a bit of a failure IMO, no matter how sweet it sounds soloed. I'm not saying it sounds bad and for some genres it's great but it's certainly less of an allrounder than I expect from a top-of-the-range P-bass.
But the most disappointing part was the lack of QC. Wires were lodged under the pickguard which is permanently disfigured and bulging outwards. There are also several dents in the upper edge of the rosewood fretboard from 1st-3rd fret, giving it a serrated look/feel. Extremely disappointing in this price range. But no worse than I've come to expect from Fender QC lately...
On the positive side I absolutely love the neck profile.