Best bass in $700-800 range.

[Pep]

I agree pretty much with what everyone has recommended. I'm going to recommend another brand some may "rag on" but I've two of them for about 2 years and they have been flawless, the workmanship is extremely good, they sound awesome and they are very affordable. Vintage Brand basses designed by Trevor Wilkinson.
I know opinions are subjective but I believe they're as good, if not better than the MIM Fenders.

 

[DarkCherry]

G&L

 

[cchorney]

Shouldn't it really depend on what sound you want, your willingness to shop on the used vs new market, and other factors?

You've already said you like the necks on the Fender P basses. That's an important first step.

How does it sound? figure out - by trying them - whether you like the sound of a single split pickup (P bass style), a pair of single coils (J bass style), a single humbucker (MM Sterling style), or double humbucker (MM Bongo style).

Do you want active (which gives a little more bass/mid/treble control at the bass, uses a battery) or passive (simple tone control, no battery to replace).

Scale length - something in a "full" scale at 34", or something shorter?

How does it feels when you are standing and playing? I loved Steinbergers until I put a strap on one and stood up, it just centered wrong for me. And basses with short top horns have more neck dive than I like.

I would worry less about name and where it was made, especially if you are buying used. Just make sure the price is fair (check closed ebay listings or what GC is selling them for used).