Rex Brown is one my favorites! Big influence on me when I started out. With all the gear you have, I am going to suggest taking out the Sansamp and the compression pedal. Focus solely on your bass, 3 PRO, and the cabinet. Next thing, don't try to dial in exactly what Rex Brown was doing on that track. That recorded tone worked for THAT MIX. Not your band situation, or what your bedroom sounds like. You will not achieve what he got, but you can definitely get close to it.
I am taking a guess you have a bunch of gear connected, and the resulting tone is less than stellar. In addition, the Sansamp is likely scooping out a lot of midrange and you are having a hard time hearing yourself at shows. Am I on the right track? Here are some suggestions to fix that:
3PRO - Don't use the Ultra Low button, Set the Gain knob so the red light flashes often. Put all knobs at noon, EQ sliders should be in the middle. Turn the Tube Gain knob to noon or all the way to the right. Start boosting around 1khz to 3khz for the high end (just a little) and see what sounds you can achieve.
Your bass - Don't boost the low end or high end, put those knobs in the middle. Turn your volume all the way up. Turn the pickup balancer to whatever sounds good to you and start playing.
The key - Get your bass tone and the simple settings sounding close to what you want, THEN add more gear to it. Many players, including the past version of me, would simply start combining effects and wonder why it sounded so terrible. Too often, none of us try to learn how each piece of gear works INDIVIDUALLY. That is the key to everything.
To my ears, Rex's tone is coming from a Stingray 4 or 5, roundwound strings, pick, and a small amount of overdrive. Dial the clean version of this tone with just the amp and bass - you will be on your way from there.
Once the amp and bass are sounding good, then add the Sansamp (I am assuming the Bass Driver DI). The Sansamp is powerful and can take over the sound of anything you plug into. If you lower the Blend knob to the 9 o'clock position and simply dial in some overdrive, it will be much more usable in achieving Rex's tone. Higher Blend settings will give you a more scooped tone. Any time you raise the Bass or Treble knobs on that pedal, the midrange response will decrease.
What Compressor pedal are you using? What are the settings? I will recommend keeping this pedal out of the loop right now.
Do you have the SVT 410hlf or the Heritage version of that cabinet? If you have the SVT version, roll the slider on the 3 PRO at 33hz down about 25%. This will cut some of the extreme low end and help you hear yourself.
Start there and report back.
Some helpful advice:
SVT 3 Pro settings tutorial thread - please contribute your knowledge!