It can be done, but you will have to flush not only the warm water in the bowl, but also the warm water in the tank. Then, you would need to flush one more time to get cool water into the tank, and them flush one more time to transfer the cold water to the bowl. So far, you will have wasted approximately 8 gallons of precious California water, and need to find a way to either dump the water into a drain line, or channel it to a useful destination.
Have you considered installing a PVC line 2'-3' below ground, where it would remain cool? Any water from this line would be quite a bit cooler than sun soaked pipes, and could be easily transfered into a bowl containing a float?
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