Reagent grade water is overkill for a watercooling setup.
Pretty much nothing in a watercooling loop is perfectly clean. Even if you rinse/clean everything you will still have softeners slowly leaking out of the tubing into the water.
So the liquid in your watercooling setup is not going to stay electrochemically inert for a very long time, no matter what you do or what type of water you start with. That is why most people use killcoils or additives - to prevent growth of bacteria and algae, and to slow/stop electrochemical corrosion (though that is not a problem if you don't mix metals in the loop - i.e. aluminium reservoir and copper coolers).
The reagent grade water will not damage the components of your watercooling setup, the worst case i can imagine would be "washing out" the softeners in the nonmetallic components quicker.