Well, I've managed to find something that would cause the system to run unstable enough for an application crash or complete system freeze. Funnily enough, it's Batman Arkham Knight.
So, setting every graphic setting as high it could go, enabling all of the nVIDIA Gameworks settings and running at 3440x1440 at 100Hz with GSync enabled, I could either force it to cause an application crash or total system lockup when running the XMP profile of my RAM. Interestingly enough, setting it back to stock would enable the game to run glitch free. I'm about to set things back to XMP to see it isn't just coincidence.
I haven't run Arkham Knight before I swapped out the old Thermaltake 1200W to the new shiny EVGA 850W P2, but I didn't experience any issues running 3DMark and a couple of other games when using XMP with the Toughpower.
I did have an issue trying to run the USB3 hub my monitor provides when using the Toughpower (it'd always report that a USB device failed, regardless of what port I'd use), but I no longer have that issue when I use the EVGA. As for USB devices, I have a Logitech gaming keyboard and mouse, 720P webcam, Steam controller dongle, Xbox One controller dongle, Logitech wireless headset dongle, sometimes a Saitek X52 HOTAS (only plugged in when I use it), the monitor's USB3 hub, and an internal USB hub as I have front port USB 2.0 ports and an NZXT cooler, but only one USB2 header on the motherboard.
So... in summary:
Random lockups seem to be related to XMP, and running stock seems to fix stability. I'm about to reboot and verify that.
System seemed more stable when running XMP and a different power supply. Again, I'll need to verify that. The power supply used could be a red herring.
Other than the wattage outputted, the main difference between the two power supplies is the EVGA has a single beefy +12V rail (70.8A, 849.6W) and the older 1200W Toughpower has multiple +12V rails (20A on the first +12V, 36.0A on the second +12V, 30A on the +3.3V, 600W combined for the lot, and the same for the second set).
Edit: Okay, can confirm that setting the XMP profile causes Batman Arkham Knight to crash. What little of it that runs also seems to indicate that there's some performance loss as well.
I'll run a few other things to see if anything else causes issues as well. If I get a chance to, I might also see how it goes when using the old Toughpower later this week or next weekend.