In the bridge pickup I like punchy power-chords and ripping pinch harmonics and in the neck pickup I like a fat sounding vibrato, smooth speed playing and clear chords. What Pickup set out of The Mule and the Cold Sweat would you recommend?
This sounds much more like the VHII's than Mules or Cold Sweats.
I've just put a VHII set in my Les Paul Custom. The seem to get ignored somewhat because of the Van Halen reference, but they're in no way a one trick pony that just nails one tone. The bridge is punchy sounding with good top end clarity but also still sounds big with it. The neck has a good mix of warmth and definition. You can definitely shred on it too. The VHII will do all the classic rock sounds easily, but has a little bit more output than a Mule. It has great cleans too.
The Cold Sweat bridge is hotter again than the VHII and will be easier to grab pinch harmonics because of that, but is more modern sounding, is tighter and has more top end and bottom end so sounds a bit more scooped. It can also sound quite bright in the wrong guitar and the clean tones aren't anything that special. I've never tried the neck pickup, but people seem to love it for shred and fluid tones.
For what you want I think VHII's would work really well.