There have been several documented cases of dyed BTAs and LTAs and H. Crispa anemones on this forum. I can't recall one for carpets though.
A dyed anemone has basically the same odds as a bleached anemone, as they are virtually the same. The original anemone is exposed to some conditions that cause it to expel all it's internal coloring. The remaining "color" is white or clear. The anemone is then placed in water with a heavy amount of dye (probably food dye) and allowed to do internal water changes for a few days or so. The end result is an anemone with no actual color full of colored water and dye-stained cells. These anemones, if cared for soon enough/well enough, can recover, but the original color will return and the attractive, dyed color will gradually fade out.