A satin stitch will give you a finished looking edge. However, that would be quite a challenge given all those curves. Satin stitching looks better when it's in a straight line.
Unless you are expert at satin stitching around curves and are consistent, I'd practice and practice and practice satin stitching around circles until you're satisfied you can do them "seamlessly."
I've used satin stitching around curves before and though I'm getting better, I'm not always happy with the finished product.
Other than that, I would have to agree with the bias binding suggestion. That won't be a picnic either, but you'll get a beautiful finish.