After an evening of drying, here are the results of my dye tests so far:
The panel on the bottom is the white fabric dyed in the "wine" color. The fabric in the middle is white fabric dunked in "scarlet". The fabric on the right was dunked once in scarlet, let to dry overnight, then dunked in wine the next day.
That last piece is the one I'm going to use. It's very nice. I'm sure it's not an exact color match, but what I was going of was a dark red, and that is certainly it. Either in display or actual use, this part of the cape will not be seen at all, so I'm not too concerned about making it a perfect match. A deep red will be totally great.
So this officially ends my foray into dying for the stabilization panel. Seeing as how I'm still only building up my prototype cape, I'll have to repeat the process soon, but I feel that I've gotten it pretty well nailed down.