Just to put in my £0.02's worth: I recently also bought a Fender Stu Ham Urge I but a Mexican one, with only the JJ pup configuration. I played it through my Trace practice combo and the Trace 8x10 at a rehearsal. Now, to my ears, although the neck pup is nice and deep, it doesn't sound like my squier P. The deepness achieved can probably be put down to it's position, which is probably the same as yours, as the two pups are quite far apart. I thought the preamp was OK, actually. Perhaps I just have low expectations, playing mainly passives: the EMG's in the Jazz don't really count as "active", I don't think.
They obviously saved a few pennies by just not including a P pup and keeping the neck J in the same position.
Can't compare them ATM, as the Urge is in the care of the Bass Centre, getting a set up and new strings. May get back to you once that's done. Might even try it against my JV Squier J and P's as a comparison.
As an aside, have you noticed that the width at the nut is quite wide, like a P? Not sure I like this, as I have small hands. This is the reason the J is my main bass, over the P. I had a Fender JP-90 until a short time ago and the nut width was more J but perhaps even slimmer. It had a similar headstock to the Urge, so I thought it would have the same nut width and it was an unpleasant surprise when I realised. So, who knows? May be moving it on.