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Chlorine is chlorine, it doesn't matter where it comes from. What does matter is that dichlor contains additional things that you don't usually want added to your pool, while bleach contains nothing but chlorine and a tiny amount of salt. Pool stores often object to bleach because they don't sell it.
WIth a SWG and a FC level of zero, you should check your PH, CYA, and salt levels, make sure the SWG is turned on, and shock the pool. Shocking is best done with bleach (unless your CYA level is too low).