What about buying one of the souvenir bottle straps at the park? Couldn't they be used on most models? Ours came with a strap. We also alternate with the mini-mister. I know they are a pain, but in the heat of the day in the summer, I wouldn't be caught without one. Then, when things cool of -- it goes into the locker with any of our other junk.
We also have 2 mini misters that we keep on hand or in the fanny pack. They work just as well -- but you do have to refill water more often. Only a problem in the summer when the heat is truly oppresive.
If you don't want to lug one, just go up to where they sell them. They usually have them in a bucket with ice and water in one. You can take a "test" squirt. Cheaper that way, too.