I can't offer a direct P pickup comparison, but my Jake has the Delanos, and while with the tone rolled off it nails the P tone IMO, it's pretty aggressive. Dial the tone back in and it growls beautifully, so that's probably not what you're after. Output is massive, my Elwood has the Haussels through a glock preamp, and the output from my passive Jake is much more brutal, add in the Humbucker I have in the bridge and it's pure rawk (which is not what I do, but it's nice to have... gotta keep these folky/country types on their toes!).
I can't speak for the Haussel P type pickup, but I find the J types pretty authentic, warm and vintagey, what I'd expect from a jazz. If that extends across their range it might be an option. I'm sure Adrian could advise.
Playability wise, I had a Lakland 55-01 alongside my Elwood and found the Elwood much easier to get around on, Some of this would be down to losing an inch off the scale length, as I spend a the bulk of my time at the deep end, the profiles are very similar. I sold it when I ordered the Jake. I'd previously played a standard Jake, and got on with it fine so that's what I went with for mine. It's maybe slightly deeper than the Lakland, but feels more rounded somehow as you move up the neck where the Lakland flattens off more, dead easy to play and certainly not what I'd call big. Compared to the few Fender's I've played, there's much less wood, and the gentler radius and rounded heel makes all the difference to me. I'm no expert on Fenders mind, I've never lived with one, so my experience is limited to little jams on other people's instruments, but they feel a bit primitive to me, and I don't get that from the Jake.
Just as a caveat, I'm talking about 5 strings here. The only Jake 4 I've ever had a little play on was Sibob's Beautiful gold, chambered beastie, at last years Herts Bash. That was a lovely instrument to play too.