There may be some larger gel packs that could be put in the freezer and placed on the back of the neck. There was a local TT in NorCal that was very hot. Pantyhose filled with ice around the neck and down the back did the trick for warmup. However, melting ice in the gym and filling pantyhose with ice
may have their own "issues".
How about a large 24" box fan that is generally pretty quiet and moves a lot of air? It seems more reliable than cooling vests / gel packs or dripping water from meting ice.