Can you tell me about the Fender American FSR Jazz basses available?

"FSR" means Factory Special Run, so it's not A model, it's a designation for for instruments made in a limited production batch. A consortium of small dealers can go together and order something different from the stock offerings for example. Could be anything from a stock model with a finish not offered on that model (e.g. a Vintage bass in Candy Apple Red) to pretty well customized like a Precision body, Jazz neck, Buddy Guy polka dots, and active electronics.

Without knowing WHICH FSR it was, kinda hard to tell.

 

Most FSR Fenders are not ordered from dealers - although that happens - but made by Fender for what reasons ever.
I remember two different Fender FSR 70's Jazz Basses. One from 2002 or 2003. This was identical with the Vintage Series 75 Jazz bass Fender offered until the new AVS was introduced a few years ago.
The only difference were the colors and the matching headstock in the same color.
Many FSR Fenders are this way. I have a FSR '64 Jazz and it is in Daphne Blue and has a matching headstock. Otherwise it is identical with the AVS '64 Jazz.
IIRC there was a US '70s Fender Jazz Special Run (FSR) about 2 or 3 years ago. This was a bit different.
The new AVS '70s Fender model is a copy of a '74 Jazz Bass. This has replaced the '75 model and doesn't have the 3 bolt neckplate with the adjustment system, but an ordinary 4 bolt neck plate.
The FSR was like the older '75 model. Maybe they had 3 bolt bodies and necks lying around and wanted to get rid of them and turn them into something special.
I don't think that the necks changed or the frets they're using.

 

If that particular FSR Fender really did it for you, why not just buy that particular bass?

 

I was about to write the same thing.

 

Model Name: Fender FSR '75 Jazz Bass
Model Number: 017-4300-(809)
Body: Alder
Neck: Maple
Fingerboard: Rosewood, (7.25" Radius/184 mm)
No. of Frets: 20 Vintage Size
Scale Length: 34" (864 mm)
Width @ Nut: 1.500" (38 mm)
Hardware: Chrome
Machine Heads: Standard
Bridge: Vintage '70s with Single Groove Saddles
Pickguard: 3-Ply White (I think it's closer to parchment)
Pickups: 2 Vintage Jazz Bass Single-Coil Pickups
Controls: Volume 1. (Neck Pickup), Volume 2. (Bridge Pickup), Master Tone
Colors: (809) Candy Apple Red
Unique Features: Matching Painted Headstock,
White Binding with Pearloid Block Position Inlays,
"Bullet" Truss Rod Adjustment,
3-Bolt Neck Plate with Tilt Adjustment Feature
Source: CORONA (US)
Accessories: Black Tolex Case
U.S. MSRP: $2,199.99

SOLD - 2002 Fender USA FSR 75RI Jazz Bass- Candy Apple Red- Matching Headstock- 1 of 250