Interesting question here... I've not had that problem given that the floor of my grow closet is pretty cool (uninsulated floor over a dirt floor basement)... my tent is a different matter and the water in THOSE buckets is a little warmer but not much - again the result, I'm sure, of very little insulation in the floor... but in thinking about your problem, I've got an idea... Why don't you mix up a separate gallon of nutes making sure it is as precisely the ppm of your bucket as possible and as precisely the same pH as your bucket as possible - and keep it in your refrigerator? Once or twice a day (depending on how quickly the bucket warms up), draw off a gallon of water from your bucket and replace it with what you've stored in your refrigerator? You'd have to check the pH of the cooled solution just to make sure it hasn't changed.. then put the gallon you've drawn out in the fridge for the next time the bucket needs cooling off... ??? It's probably a whackanoodle idea and someone might come along and tell you why you SHOULDN'T do this - but it's about all I can come up with at the moment... except for the idea of painting your bucket white... dark colors absorb heat, light colors reflect it.... They make white duct tape, too, if you don't want to paint, you could just plaster white duct tape on the top..
Good luck... you're right, you need to cool it down to at LEAST 22C!