A ton of compression - a dual band one may work so you can squeeze the highs more than the lows.
A Tube Screamer or clone for a bit of grit - I'm using a Digitech Bad Monkey which does the job nicely. Whilst it give a warm, natural overdrive it also retains the clarity of the clean sound without the need for a blend.
Set the amp quite flat, boost the mids if necessary and dial out the very top end to remove the 'zing'.
The key though really is the 2 signal from the 2 pickups. You could try splitting the signal from your bass and process them individually, but the Attitude's p/up configuration really does it a lot better.
I'm pretty much nailing the tone on a relative budget - relative being compared to Billy's rig.
I send the P p/up of my Attitude through a Behringer eq to dial out the low end to get a twangy guitar tone, then through the Bad Monkey for grit, then into one side of a TC Nova Dynamics compressor set with fast quite extreme compression.
The 'woofer' neck p/up goes straight into the other side of the TC compressor set with a slower attack and mild compression. I then combine both signals via a Boss LS-2 and that goes straight into my LMII head, on which the eq is pretty much flat, but using the VLE filter to roll off the very top end and inject a bit of warmth.
Like I say, on a regular single output bass, you could split the signal into my above signal paths which could work quite well, although you may have to also eq the 'woofer' side to get a fuller bassier tone.