Well, I guess that would work,with a few small chicks or such. But if I was going to do that, I'd fill, freeze, and rotate a set of the containers themselves, that you screw the tops onto, instead of the tops. Another possilbity if your freezer space is limited...just rotate stacks of regular ice trays, dumping the cubes into those containers and finish filling with cool water before screwing the tops on. Bear in mind, the smaller the water/ice volume, the faster it will all melt.
And as others here note, neither misters nor evaporate coolers, swamp coolers, as of much use in humid climates. That is what did in a little flurry of fledgling small commercial rabbit industry down here a few decades ago...standard for that kind of operation in most the country are long narrow barns, with one end covered with a big evaporative cooling pad and pumping system, and huge exhaust fans at the other end, to draw cooled air in through the wet pads and across the cages of rabbits the entiire length of the building. It just didn'twork in this humid climate. In a consistently dry climate, they are fantastic, and much cheaper than air conditioning. I visit relatives that live out west of here in Texas, in the Texas hill country, Kerreville, Bandera, CampWood....and they use them in their homes, they cool as well as any AC we use here. Another difference, try to use them here, in humid climate, you usualy get mold problems, big time, but out there, it is so dry, the added moisture in the air is actually good for you, helps with excessive drying of skin and even mucous membranes as you breathe.
I hadn't thought of that! I've been wondering how I'm going to keep my chickens cool this summer. I'm in south Louisiana and have weather like yours. Summer should be starting in just a few weeks around here, up into the high 80's probably in a month. Judging by the extremely mild winter we've had, I'm counting on a miserable summer.
So, you keep containers of ice rotating in your freezer to put out for drinking, wading, sitting on, etc...? What do you think about keeping a couple of waterer tops (the plastic ones that screw onto the resevoir kind) rotating in the freezer? The ones I have are not quite half a gallon each, but I have 4 total and could rotate that twice a day. As in set one out to drink as it melts and have the other two ready to go for mid-day or afternoon while refreezing the first two. I don't have a lot of freezer space either so I don't know that I could keep a lot on hand but I surely want to keep them as comfortable as possible.