This is how I would do it. Bias binding would be so........ much easier. I would slightly (just slightly) tug on the binding as you sew it on. Turn and handstitch down.
Others will have other ideas also. Happy sewing!
Oops Raptureready and I are on opposite sides when it comes to tugging on the binding. Maybe tugging on the binding is the wrong word here. I find when I stitch a circular item like a small round that if I just pull a bit on the binding as I sew it lays better for me.