Hi and welcome!
I also have a S570, but I made mine into a classic rock & metal axe that can go into modern stuff too with a RY bridge, IT middle and Mule neck. Love that combo. The RY deffo puts it on the bright side, but that is what I wanted for that guitar. Very versatile now and I just love playing it.
For you I think the HD would work very well. It is full, smooth and warm, with character. Rounded top in a musical, but never dull way. Very versatile PU.
As for the middle either IT or SH should work nicely imho. The IT is a hotter, fuller, dirtier take on a classic SC and the SH is the same but even one more step up.
For the neck, yes the CS would work, but depending on where you really wanne take that the Emerald might be a better option. It is more organic and lively than the CS, which makes for a combo that has been recommended awfully often here recently. The CS pairs beautifully with the A-bomb, but for the HD I recond the Emerald is more on the money.
In regards to the Warpig I cannot say much. Could work, but I dunno, it really, really is a beast. If the guitar is very bright, maybe, but otherwise I would be more carefull. Not to paint it too dire, but due to the bottom end it is more picky than most PUs