I use the frio wallet too, you soak it in water and the gel inside swells up. Then as the water evaporates it cools the wallet. I find I need to top up the water every couple of days and you do need to keep them somewhere where the water can evalorate so not in a fridge or cool box.
I usually end up taking a clip lock plastic tub with my spare needles, testing strips, jelly babies etc in it. I can usually find somewhere out of the children's reach to store it. Even the back of the car or locked in the glove box works.
Maybe giving the 2 year old her own bag to carry would help you keep yours private.