you might try facing the edge instead of binding or using rattail cord. (Using rattail makes the edge a little stiff which you might not want on a baby quilt.)
Facing a quilt edge is just like facing a neckline in a garment - make the strip wide enough to stitch to the front, turn to the back and still have an inch or more on the back. This leaves a knife edge on the quilt - not a real binding, but covers a lot of area in the back. Plenty of videos on line about facing a quilt (or garment neckline.