First I soaked the damaged pieces in hot water for about 1-2 hours, minus any suspect pieces that could have caused the dye transfer stains. I added a generous amount of Vanish Napisan to the soak.
Then I placed them in the washing machine to wash again. I set the washing machine to "Whites", which enabled the wash cycle to operate at 95C (normal cycle is 40C on my machine). Again, I used a generous amount of Vanish Napisan. It worked! All the stains were gone from every piece, even from those that looked hopelessly unrescue-able.
Pictured here: Napisan Vanish and some white clothes after rescue)