... and apparently it had mysteriously fixed itself in the process of being emptied. As soon as I powered the system up I could hear the pump humming as it sucked air. I could feel it vibrating to the touch as well - something that it had not been doing when it was in the case. I fetched the video card again and snapped the hoses together to confirm that it was working, and the pump obligingly started pushing air through the lines. I popped the cap and refilled the fluid reservoir, and after a lot of alarming gurgling and labouring, it went right back to work as if nothing had ever happened.
It is back in the case again, and the GPU is running at its normal idling temperature of 54C. I have no idea what happened, but I am going to keep a closer eye on it until I feel that I can trust it again. I am not too concerned about it overheating when I am not here; my power settings shut off the display after 20 minutes, and the card doesn't generate a lot of heat when it is idling. Still, this is the worst kind of component failure. I would rather have something break outright so that I can send it back for a replacement. If it cuts out again I will send it back regardless and advise them that, whether it works or not on arrival, it had quit working twice in my system.
Hopefully I don't need to do that.