White is just traditional, not mandatory. Have you looked at DWR on Pinterest for ideas? There are many non-traditional quilts shown. "It's not you grandmother's DWR anymore."
With that being said, I think you just need to lighten up the rings. Give them more contrast. (I think) your darker reds are blending into the print. (That is MY opinion. It might be how the colors show on my computer.) I have also seen the rings done with piping on the outside of the rings (both sides of rings). Maybe do a thin black flange outlining the rings. Ohhhhhh, then you could even sew the rings down onto a solid piece of the print, in the ditch of the flange, and not even have to piece the curves. (Sorry, my mind got out of control.)