KoolAid, I have read, works only on protein fibers, which is why it can be used to dye hair and wool. Cotton, I think, would be considered more of a starch, like wood.
I don't know how dying cotton ease would work. Seems like the synthetic part might not take the dye as well so you could probably dye it but may not be able to end up with a very vibrant color.
Interesting to read about the Rit. I dyed an Ikea chair cover with purple rit and it came out looking great but after one wash it is faded already somewhat. Fortunately you don't have to wash chair covers too often.