Yes, it's a Fender Jazz. I'm afraid I don't quite understand the second half of that first sentence, but I love the sound of my J-bass too!
The little trick: You must have noticed how stacatto the bass notes are played. We sort of wanted to make my bass part approach it more like a Stand-up bass. After a little experimentation, we ended up shoving a bunch of those round cymbal stand felts under the strings and up against the bridge! There was enough in there that it actually changed my string length and I had to re-tune! But it made the strings really, really dead; no sustain. When I played, I didn't lift my left hand off the fret, or kill the vibration with my right, it just sounded that way. After that, we triple tracked it. It doesn't sound like a stand-up really, but I think it sounds pretty cool.