Yes, you are correct. Your thinking is very good and on the right track.
I imagine it must sound like a broken record sometimes. Bleach, bleach, bleach. But it's the safest thing to add when chlorine is needed. It just adds chlorine, water and a minute amount of salt. Until new posters come back with water test results, liquid chlorine (bleach) is recommended for just that reason.
Then there's situations like yours. Liquid chlorine then becomes the only chlorine source that should be used. Looks like you done your homework.
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