Blah blah blah. I have $1,000. I play a 500watt GK amp and I don't plan on buying a new amp for a while.

Basses I'm looking at:
Sterling Ray34
Fender Geddy Lee Jazz
Fender Aerodyne Jazz
Gibson SG Faded
Warwick Standard Streamer
Jackson CMG Concert
Godin Shifter (Looks insanely badass)

Which one will let me get the best bang for my buck?
Are you willing to go used? Because you could skip the Sterling Ray and easily get a used MM Stingray with that much.

You really need to go try some of these, you're comparing apples to oranges to bananas here. The Geddy Lee has an insanely thin neck, while the Aerodyne has a P/J pup configuration, the Gibson SG Bass is a short scale, etc, etc.
Composite Aficionado

Spector and Markbass
Yeah, I'd go used.
And I'm probably going to try them all out, but I just don't want to go through a number of basses I might not like at all.
Currently, I play in a Hardcore band. (i.e. Down To Nothing, Have Heart, Bane, Cruel Hand, Madball) I've noticed most bassists play P-Basses or Jazz basses. I'm not big on the big P-Bass body, I have a lot of crazy stage antics and I can't have my bass weigh me down. That's kind of why I'm looking at the SG Bass, plus I love Mike Watt's work. I just don't know how they can handle distortion or drop tuning.
^ A Sterling or a MM would be heavy. The aerodyne is lite. The Godin is an exceptional bass. Though heavy, the MM's are built great and sound great too. I would recommend a used MiA jazz.
<Insert Witty Comment Here>

1981 Fender Lead I Seymour Duncan humbucker, Mesa BoogieIIIRectifer, MKIIRhodes,PRS
If you are going used, follow Tostito's advice and get the MM. If not, I think the Ray is the best of all the options you provided.
I have the warwick Pro series corvette. And its ****ing heavy. It hurts after a few mins. If you are going for a guitar with heavier wood, or a larger body, keep in mind the weight, as you said. If you are going to be running around, it makes a hell of a differece.

If you get a heavy as **** bass, you could always exersize a little and be a hard-arse XD