Doesn't a P/J sound close to a P when the P pickup is on by itself?
Definitely, although on most PJs, the P pickup is are a little closer to the neck than the purists "sweet spot". There is an easily heard change in tone by moving a pickup an inch or 2. My Grabber proves that. But a significant difference in a mix? I really don't think so. My PJ sounds like a P when the P is solo'd.
I voted other, though, because I just LOVE my Aerodyne (PJ) now that I have it tricked out the way I want. The original P sounded like a P, although the J pickup was noisy and anemic relative to the P. EMG Geezers definitely have a great vintage P tone and both the P the J pickups are beefier now, the J much more so, and growlier. (Spell check doesn't think "growlier" is a word but I know you know what I mean.)
I have a Lakland J that is great for some stuff that I do, and I always enjoy playing it. But if I had to stick with one, it would definitely be the PJ because of the range of tones. I pretty much always have the P dimed, so tonally I like the P a little better than the J. P tone fits a lot of what we're playing these days. I'm not married to Fender, but the 'Dyne is a great bass and I'll check some out when it's time to expand the stable.
I definitely prefer the J neck profile over the P. Never a fan of chunky necks myself. That's completely personal taste, I have nothing but respect for those who can get music out of a baseball bat. But I'm much more comfortable with a thinner, narrower neck profile.
Next up is a 5er. It'll definitely have something approximating a J-type neck. Not sure what brand -- maybe a Stingray or L2500. Could be a PJ or a J. Probably won't be "just" a P.