When we were tent camping I bought one of those mister kits, same brand and size, and tried to hook it up to a garden pump sprayer.
Never worked, but I still have the mister kit so I might steal your idea.
And, at that time, we found some hiking personal misters that worked fantastically well. Just a bottle about the size of a big coffee mug with a pump and then a hose with a single mister head on it that you clipped on your collar. When you got hot you opened the valve and misted yourself in front of your face.
We have been using the kit like yours in the backyard on a popup tent and it works great. the nozzles do plug up and begin to sputter. We usually put the nozzle into a cup with CLR in it overnight and it is ok and producing a fine mist the next day. CLR is that calcium-lime-rust product that they show on tv getting rid of calcium deposits on your showerhead.