I'm not sure if this will work for you but one idea we use at work (outside in AZ temperatues 90-100+ wearing bullet proof vests) is to slip those reusable freezer packs between our undershirts and our body armor. It does a decent job of keeping the core cooled. That said, it does not last long but it is definitely better than nothing. Perhaps you could put one under the weight vest?
"Corn syrup is everywhere; check your pockets." - Sh*t Paleo People Say
"This really steams my monocle!" - RichMahogany