Unfortunately, I don't think either of those ideas will work well. Dry ice doesn't like to make good contact with other solids because it's constantly changing from solid to gas. There wouldn't be much thermal transfer, and if there was, it would freeze the water and crack your rad (Dry ice is ~75*C).
If you just put your rad in a bowl of water or ice water, the thermal transfer would be much better, but there would be poor circulation of cold water through the hot rad. The water right around the rad would heat up and still not cool well.
I don't know if you have a prebuilt loop or it's one you made yourself, but if you want to test whether it's worth it to make a better loop, pull the rad off completely and put the two loose ends into a bucket of ice water (put a filter of sorts over the inlet) and then you will have a short period of time to run your loop at near 0*C. Eventually though, the loop will heat up as the ice melts.
I'm sure others will chime in with better ideas, but that's what comes to my mind.