OK, I just watched the clip- notice how he's playing with a pretty aggressive/dynamic technique back between the pickup and the bridge? Then notice how the bass tone has hardly any dynamic qualities whatsoever, it's just solid and thick and steady-level? With that playing technique his sound should be super bright and jumping all over the place volume-wise. I'm betting he's going through
at least two compressors.

No, really, I mean at least two. Plus a hint of chorus.