Maybe it's just me, and I have to preface this by saying that Randy's untilmely death and his incredible playing were what made me know I wanted to play lead guitar. BUT, I think his tone, or at least the portion that came from his pickups, wasn't all that groundbreakingly unique. Most of his best solos (Mr Crowley, etc) were recorded on a stock Les Paul custom with stock Gibson pickups. His studio tone was the result of putting a Marshall halfstack at the bottom of a concrete stairwell and playing from the top of the stairs, and then rerecording and layering the same parts for a big thick tone. His live solos were just the brute force of a Paul going through a Marshall halfstack with a couple light effects and OD. That's just it, there's really not a huge amount of magic, aside from his soulful playing which most certainly was magic.
I'm still shopping my first set of BKP's, but I'd side with most of what I'm hearing here, something that approximates classic Les Paul tone, maybe with an extra little touch of gain would probably do the trick nicely. Black dogs or mules or riff raff. Or hell, I'm looking very hard at the Cold Sweat myself, what with Sykes being my other icon, and his tone on a Les Paul was no slouch either.
I think just about any hot PAF style from BKP will do a good Randy tone, in the right guitar with the right amp rig. The rest comes from a lot of talent and a lot of practice. Best wishes in that endeavor.
