Might work, but two issues would be of concern: Bleach is corrosive, so anything in contact would have to be very resistant to lye. Second, bleach degrades with sunlight and heat, so storage would have to be in a cool shaded area or you would have to add bleach to the tank daily, defeating the benefit of the system.

Might be better to fill the tank with a bleach water mix so the degradation would be slower (as I understand from posts here), and it could be mixed at such a concentration that it could run all day to add the correct amount of chlorine.