I don't know if they'll dye or not, but.....
Here's a tip from when i painted aircraft for a living. First we'd have painted it black (that would cover all the darker colors and prevented their bleeding through to the new color) then, and only then, procede to paint it the desired color.
It might work for dying fabric as well?
And you might try bleaching out as much of the original DPM as possible first.