The "distortion" sounds like a touch of fretbuzz to me (but the deliberate kind that can actually sound cool in some situations). Maybe you could try adjusting your action lower and/or tighten the truss rod a little, to the point where you get that grindy hint of fretbuzz when digging in, but can still get a clean tone when playing softly. Just make sure you dial in a relatively slim tone with enough upper mids to let the buzz/stringnoise cut through the mix. The choice of stings can make a HUGE difference too of course.
Another way could be to fret the notes VERY softly with your fretting hand. This can also create that kind of grindy buzz/stringnoise.
I might be wrong but I honestly don't hear any other "distortion" on that track? It all sounds like fretbuzz/string noise to me.
I think people are seriously overthinking this (and getting it totally wrong in the process) when they bring B15s and all kinds of expensive digital gear into the discussion. Just straight into a decent DI and a good (preferably transformer based) micpre should get you there IMO, as long as you get it right at the source.