I conducted the first test for this project idea, and it was a failure.
I used thermal adhesive from Microcenter to glue a Peltier element between a metal bowl and an upside-down computer heatsink. The leads of the Peltier were soldered directly to 12V wires from a computer power supply.
The result was very weak. Whether due to bad thermal bonding or not enough Peltier power, the device wasn't able to appreciably cool even a small amount of water. Th thermal adhesive also became very brittle on the cold side, and I was easily able to pull the bowl off with my bare hands. The bonding on the hot side is much better and stronger.
Clearly I have much more to learn about this. I plan to use 4 Peltiers, and I will try to put more pressure onto the bond to allow better heat flow.