As you know, red is a notorious bleeder. That said, with DMC floss (not sure if you're using floss or thread) I've usually not had a problem, even with red. It may have run a tad when dampened but most needlework that I've used floss (hand embroidery; needlepoint), I end up hand washing in the sink with liquid dish soap and have never had a lasting problem. Can you try, say a wet q-tip, with a touch of dish soap and see if that removes the dye before continuing?