yeah that’s it, it’s a basic lever-effect thing and I believe that’s the reason for the Precision Bass shape. because Fender design is right, right ? 
No religion here : I must say that, look-wise, a PB is not so good looking to me (I don’t like Stratocasters much either ; basicaly the guitar version of the PB).
it’s just the thing that sounds right to me. and even with this extended upper extended part (don’t find the words in english, sorry) for the strap button, a Precision has neck diving issues either. so, well, Fender design is right but sometimes it doesn’t work so well 
My guess is that all those “vintage design” basses must have this issue. The more modern body shapes often show a very extended part for the strap button and lighter tuning machines, so there is no more (or way less) lever effect. so less or no more dive.
I guess you should have the same issue with your beloved JB ? because it’s still a very vintage design. Fender design have not changed over years, just very minor updates but we still play very vintage design in fact (my PB is 99% a '57 designed one, your JB is a bit younger but only about a few years ! impressive to realize that). In my opinion we can live with it, but again that’s just my opinion. I’m OK to sacrifice some practical things to the tone and feel of an instrument.
I never asked an instrument to be perfect. I don’t like perfection very much. I like an instrument that will do ITS thing very well (you will like this thing or not = you will love or hate this instrument) and this goes with far imperfect design. because perfection is perfect but so cold !
Again you know I don’t like Gibson design very much (to be fair 
) but if it gives you the tone and feel you seek/like, that’s just great.
I highly doubt about it. I’m not an expert about Gibson bass pickups designs but I know a bit about guitar pickups, and those things are pretty much identical (except resonance frequencies, winding, pole number … very minor things … still the same design on a bass and guitar pickup, very very similar).
If you have those screws, I recommand you to rise the E pole with a screwdriver and ear if it makes a difference. I’m almost sure it will. If it doesn’t work you will just have to screw the thing back but I’m pretty confident about what I just said. (still not 100% …)
oh that’s strange, would it mean the pickup is not wax-potted ? possible …
take your time, take your time … you’ll have to record a cover anyway