When you made the skirt - did you make it with wedges like pieces of pie?? If yes, then maybe you got your measurements a little wrong and need to correct the wedges by taking a larger seam as you get closer to the center.
If you only cut a large circle (all one piece of fabric) to start with - how about this fix....do a large running stitch around the center circle (you can still do this with the layers - and then gently gather it until the skirt lays flat .... once that is done, knot the thread and then machine stitch over the gather...being sure not to move or stretch it. Then you can put your bias binding on to cover your stitches. It may be a little different than what you planned - sort of having pleats in the center where you gathered.... but your skirt will lay flat - and no one but you would ever know you didn't design it that way. I wish you very good luck, as it sounds like you put a lot of effort into this project.