I've dryed the 6 sinew bows I built. But I've never colored after I put it on the bow.
I clean the sinew as if it was ready to be but on the bow. Dyed it in warm water and rit dye the color I want. I've taken it out of the pan andlet it dry and I've but it on the bow right out of the dyes pan. But I like to let it set over night sterring it now and then. This is how to get the brightest colors.
I did it for different reasons than you I wanted to get some contrash from sinew to bow.
I dryed the sinew on osage green twice,light gray,dark gray. I even camoed a sinewed osage for a guy in CANADA. But green (SHADE UNDER HUNTER GREEN)on osage I think looks SWEETTTTTTTTTT. My favoret color on osage.
I did one really nice, I had a 2 small sidewinder skins. I cut down each side so you just had the center of the dimion. I dryed the sinew gray then put the skins center over the dryed sinew. I put on a beaver handle,you could see the gray under the edges all the way down the limbs to the handle.