The celltei bags I have all have a little mesh pocket inside to put a blue ice type thing in to keep them cool. (or a hot pack in winter to keep them warm). They also have water available in a small leakproof bottle. I really like mine. I have way too many bags and three of them are Cellteis.
Maybe you could adapt one of the bags you already have to hold a frozen blue ice thingee. If it already has an interior pocket maybe you could put one in there. Just a thought...
Basic Pak-o-Pet Water Bottle - 8 oz (No Leak, No Drip!)