Interesting problem, I had a few thoughts about this.
Have you tried playing with the pcie link state power management settings in the windows advanced power setting menu? I wonder whether setting it to off, for ac and battery could make a difference
Maybe try a different vbios for your 580m?
Are you running the latest chipset drivers for your machine?
I wonder how a different os would fare in this situation. I would try running Linux off a live USB and see what happens.
Or maybe create a partition on one of your disks, to install windows 10 on for test purposes.
You can get windows 10 licence keys off eBay for a few £ (which I've bought and activated successfully) not bad considering the MSRP of the disc sets with a key.