Kind of funny that your pool guy says that grocery store bleach, and the store's wouldn't since pool stores usually sell a strong liquid chlorine.

As stated, there is 5.25% to 6% sodium hypochloride in what is labelled as "bleach" and usually 10% in the pool stores "liquid chlorine". If anything, the pool store produce has the greater risk of bleaching. If you add it slowly in the deep end in front of an outlet, there shouldn't be any problem.

I tend to put some in the pool, then put the jug under the water and top it up with water so it is more diluted.