So the troubles seemed to be over with the ESXi server crashing when I was starting VMs that used PCI devices which had been passed through, but I was wrong.
The Win10 VM that had the GPU passed through is now once again either locking up the computer (ESXi locks up) or I get a PSOD (purple screen of death).
I read about others reporting similar issues and I updated the ESXi install to the latest build of version 6.0.0, but the problem persisted.
For reference, this is a listing of all the ESXi versions and updates and links to how to apply them (source).
One thing I just tried is disabling Interrupt Remapping (source) to see if it had any effect. It had no effect.
Removed and added the GPU device from the list of passed through devices.