First off, it sounds like there is some sort of residue coming from your water storage tank. I would probably do 2x per day 25% water changes from tap and see if it clears.
Secondly, my only experience using API water conditioner was not a good one. I don't recall it giving a bleach odor, but my water did cloud, there was a slimy buildup in the tank up and I lost some fish. I dumped it and returned to using Prime...