If I am understanding your question correctly, you want the center of the circle to be solid white and outside of the circle, white on white.
This is what I would do. I would draw the inner circle on solid white, the size you want it to be finished. I would then apply a line, about 1/4 inch wide of Liquid Stitch(fabric glue) all around the circle, over the drawn line. Let it dry. When is clear, cut on the drawn line, right through the circle of glue.
Iron the circle with the glue facing down, on to the white on white background fabric. Press with the iron. Just sit the iron on top for a few seconds without moving it to prevent distorting the circle. I guess you will later applique the dark blur ring around the solid circle, so you don't need to finish the edges you just glued to the white on white. But I also guess that you don't want two layer (solid white over white on white).
To prevent the bulk, just sew a line around the white circle with the sewing machine. Turn it upside down and now you can see the perimeter of the circle in the back of the fabric. Clip a small hole with sharp scissors on the back of the white on white, being careful not to cut the white. Make the cut close to the edge, so you can reach the edge easily. Cut right inside the circle that you just sewed removing all the excess white on white from the back. When you turn it over you will have a white circle glued to the white on white, both single layer, and you can applique the blue ring covering the exposed edges of the white.
Now, go back, read this again, and it may make sense.
Maria