Make sure to rinse your prints well after the fix and before the bleach bath. Carried over fixer of any type will accelerate the bleaching and shorten the life of the bleach solution. Ammonium thiosulfate is worse than sodium thiosulfate in this regard. If you rinse well before bleaching, say 3 minutes or so, then very little fixer gets carried over to the bleach.
FWIW, I "bleach back" prints occasionally for contrast increase or for removing paper fog. Agitation in the bleach is tricky, as uneven bleaching results if the agitation is not even. I like shorter times and very little agitation.
Have fun,
Doremus