We get our drinking water from a local artesian well and store it in 7 gallon containers for up to a month. We keep it in the basement out of the sunlight where the temp is usually around 65 degrees. The containers are filled to the top with little or no air in them. Never had a problem. I can’t imagine drinking water with bleach in it.
If for some reason bleach has to be added, for instance if the container or the water was dubious, the chlorine will evaporate in 24 hours. I used to fill gallon glass jars with tap water, put a clean kitchen towel on to keep dust and bugs out, and in 24 hours the horrible bleach taste was gone. I usually had several jars lined up every day.