If I understood you right, you have a small area on the quilt's edge where the top is a little short. If it's more than 1/4" from the batting's edge, then a 1/4" binding will not work. If it's about 1/8" from the batting's edge, then a 1/4" seam for the binding would only be catching about 1/8" of the quilt, and that could potentially fray unless well quilted in that area. You are right that you can make the binding wider if you cut it at 3". One thing to consider is whether you will lose points on your quilt blocks if the binding is wider than 1/4".
Edited to add - the binding seams generally do not line up with any seams in the quilt. There are a few very special quilts where they do.