I would go with removing the offending banner and appliqueing a slightly larger one over the top. (correction, I would applique the larger one over first, then remove the other, puckered one from the back.
The new one you put over the top could be embroidered first, with correct stabiliser behind it, and the place that did the embroidery should do that for you free of charge. Take them a sufficiently large piece of the green fabric so that they can get it in a hoop, and ask for a replacement.