You could cut your binding a bit wider, maybe 2-3/4 or 3". (Cut a couple of test bits to see how wide it will take.) Trim your quilt and stitch around the edge. Attach binding to the back. Attach the eyelet to the front, pretty pointing to the quilt. Bring the binding around the front, pulling it around the wrong edge of the eyelet.
Or take the eyelet apart, making it flat and use it like a flange.