It would appear from your procedures that you have tried at least two bleaching agents. You could now be entering an area whereby you are chemically "rotting" the cotton.
From where I am, my only advice would be to stop now, you've done enough.
My approach is to firstly protect the flooring beneath the rug (if on site) and the rug itself from the fringe treatment. I use a microsplitter in a hand sprayer with a Tee Jet. Pointing away from the rug and held at an angle, I will spray the fringes very generously. As you do this you can see the soil being released as you go. Hand (spotting) brush along the fringe towards the tips. Rinse/extract. Repeat M/S and rinse as required. Several dry only strokes. Spray with neat WHITE (DISTILLED) vinegar. Dwell for a couple of minutes or so. Dry extract, do not rinse.
Not all stains will be removed and not as white as bleaching, but cotton will not degrade with treatment. Quite often, a fringe will be an extension of the warp threads. Start damaging this and you could be in big trouble.
Safe and happy fringe cleaning

Ken