Anybody up to get me THE bass-tone?

The Fender Jazz isn't the problem.
The Saffire's instrument input isn't the problem.

The lack of new strings IS THE PROBLEM!

Just tell the band that if he doesn't put on new strings then the bass sound is going to be shit. End of story.
No amount of eq or processing can bring out clunk and aggression that doesn't exist!

Put your foot down man, they'll thank you in the end.
 
New strings are essential! Can't get a great tone without it. So cheap compared to anything else too, especially if they plan ahead and buy online in bulk before the recording.
 
Get new strings and then give your DIs to Dan Swanö.
Seriously, he has some of the sickest bass sounds ever.

I would totally do that if it was for free. But as I said before : I won't pay money for a band that's not mine. But thanks for the tip. Maybe i'll sind him the DI's of my band, when we're recording!