You plan to continuously run tap water through the tubing? Unless I misunderstand you that's going to waste a lot of water. Theirs a much better way to run it without wasting nearly that much water, and not waste much at all if you use a timer or rig up a temperature switch. The only water lost is just what's lost through evaporation. As well as be able to get the water cooler than standard tap water.
As for the steel coils. Those being for making beer, I'm sure they are food grade and would probably be fine. But I can't say for sure that they won't react with any of the mineral salts or pH of the
nutrient solution. While metal is a better conductor for heat transfer than plastic. I would personally use poly tubing. Polly tubing is standard in the geothermal industry, wont react for sure, and way cheaper. That stainless steel coil is a total of 20 feet worth of tubing for $40. While the heat transfer with poly tubing isn't as good as metal, the fact I can get 700 feet of 1/2 inch poly tuning for about $45, and much more feet for 1/4 inch tubing. That more than makes up for the slower heat transfer with way more surface contact (50-100+ times more).