You will not, and cannot, get the same sound as a Jazz out of a Precision. The only way to do that is to install two Jazz pickups. Which means you might as well use the Jazz...
Any good P pickup ought to do the job of sounding like a P. And why is output such a big deaal when you have the SABDDI there do sort out the level?
They are different animals, and there's little you can do to make them the same.
A P with a jazz bridge pickup comes close, probably the best of both worlds if you must use a a Precision Bass.
I'm not a fan of actives in any way, so can't help there. As far as I'm concerned, signal levels that high are best generated inside the amp, not smacked into the input socket.
Which is why I don'tuse a SABDDI either...if you have a good amp, you don't need one, except for overdrive.