BTW Even if it were either a heat or power supply problem, it seems rather coincidental that on the occasions that it does reboot, I happen to have hit the render button.

Not necessarily. I experienced a similar problem shortly after assembling my first Athlon rig. It appeared to run fine, except whenever I tried to use Poser. Seconds after clicking the render button, the stress that Poser places on the CPU would push it into the red, and my mainboard's thermal sensor would immediately power down the system to prevent a China Syndrome.

Turned out I'd improperly mounted the heatsink. Only Poser (and a couple of CPU torture-test apps I tried later) was able to heat the chip beyond safety parameters.

And I couldn't really check the temperature readings to confirm that it was a cooling problem, because it always happened so fast. Thankfully, my neighbor was more knowledgeable about such matters.

Download the Prime95 utility and run it on your system in torture-test mode. If it also triggers a shutdown or reboot, it's almost certainly a CPU cooling problem.