Can I lighten my hair with clarifying shampoo??

I had 2 demi-permanents put on by a professional on my blondish brown high-lighted hair(the first turned out red, the second goth black). I’ve been washing and washing most of the black is gone, now comes the red. It’s currently a reddish brown. I don’t want to strip it.

Here’s my plan: (will it work??) I bought a good clarifying shampoo, going to put it in my dry hair then dry it will the dryer, add more then keep it on for 30 minutes, wash, then keep a conditioning mask on for 30 so it will restore and dryness. Do this everyday. Will it get the dye out?
Should I add a small amount of bleach to my shampoo or would that be kind of dangerous as it could go orange??

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