Well, I guess that adapter cable might come in handy sooner than later. I hooked the controller up to the still, made sure I had the control knob turned all the way down, set the kill switch to "off", and plugged the power cord into the 240V dryer outlet. When I flipped the kill switch to "on", there was a flash of light and a loud pop. It also tripped the breaker to the dryer circuit. I don't see any charred wires or melted insulation, so I'm pretty sure it probably fried something in the Chinese controller box.
And now comes the depressing task of pulling it apart to figure out how I screwed it up. I think I'll buy separate components to repair it, instead of buying another of the pre-assembled boxes. If it happens again, it will be much cheaper to buy a new SSR.