Harmony - glad you could find some sort of solution for the dye problem.
Ignore this opinion if you wish - but I'd be tempted to just use the red fabric in red themed settings - just in case. I once had red striped flanelette PJ's that gave me cause to wash red brown purple things together. I doubt they stopped leaking colour for a lot of washes.
When I made dgd's cotton crochet quilt I wanted a different brown and dyed the brown squares. Also a few other things I thought I would like to have brown. I now have 2 pairs of light orange socks. Just lovely. Not a great deal they match though.