I think you can run any program you like. Whether or not it will be optimal for you, or be optimal for achieving your goal total is another matter. The same applies to variations.
I know it's probably not quite the answer you wanted. Greyskull is a good program. It isn't really a PL program, but from what I know of it you would probably get some decent progress on the comp lifts out of it. Hell, 5/3/1 isn't a PL program and I posted my best total after seven months on it. Anything that has you hitting your squat once or twice a week, bench once or twice and DL once or twice should be fine. Ideally you'll squat and bench twice, for 531 that just means benching as a press assistance and squatting as DL assistance. Not sure how you'd work it into Greyskull.
If I was going to pick a program in your case, I'd go with Cube. You get to use bench and DL variations, there is a bodybuilding day and it's a dyed in the wool PL program.
The variations are another matter. They're fun. They do, to an extent, target weak points. They're useful if you're hurt. Apart from that, though, they just aren't quite as good as the comp lifts for getting better at the comp lifts. I think for equipped lifters and very advanced lifters it is a different story. For us, not so much. If you're set on incorporating variations into your training I could only recommend paused squats and bench; paused front squats (amazing DL builder); and snatch grip DL. Honourable mention goes to cambered safety bar squats, but they probably only are worthwhile if you tend to fall forward when squatting.