I am sort of new to this replying but I thought I would add my two cents worth. I work in an embroidery shop and I think your challenge here is embroidery on two layers of fabric. When she put this piece on the machine, she had to use a hoop and that hoop stretches the foundation and keeps it taut for the embroidery. When we add an applique, that piece might be ever so slightly off in the grain and once the machine started stitching, the letters will pull that fabric that is not hooped and hence the puckers. I tell quilters who are planning on embroidery on a quilt piece to mark the pattern on the top of their fabric and leave me a generous margin so that I can hoop it. Then I give them back the piece with the backing on it for them to applique it. I don't recommend their doing the embroidery directly on the quilt. While it is certainly not their best work, many people who do embroidery for a living do not have a clue about what quilting is and they probably think it is okay. Professional embroidery and quilting are really two different things and not every shop has somebody who understands the difference.
Your embroidery can be removed and by whoever did it. They should have a small shaver that they will use on the back side and cut through the bobbin threads and then pick out the top. I would then remove the banner, have them do the embroidery on just a single layer of fabric and then applique it down. The foundation should not be compromised and your finished product will be something that you will be much happier with. I hope this helps your decision.