From what I can tell from the pics, your tension looks good so that shouldn't be the issue. I do agree with nativetexan that some of the stitches look a bit large but it is hard to say for sure because your pics are so close up. A few loose threads (unburied tails or not locked in place and cut) are all depending what your customer wanted. I have heard of some who want the tails left so they can knot and bury them. I won't bury threads for a client, (heck I rarely even bury my own!) if they want the threads buried I will leave the long tails for the customer to knot and bury. If they don't want to do that (which I have never had that request) I lock my threads with several tiny stitches in the most inconspicous place possible, like in a ditch or an area where many cross overs have thread build up anyway so my tiny locking stitches aren't obvious. I then bring both threads to the top, test them with a little tug and snip them. Loose threads left from piecing I do snip off as I go but sometimes a few are missed. But loose tails from the quilting do not cause issues with the quilts integrity as long as they are locked and won't take out several stitches of quilting if you cut them. If they aren't locked in place that may be what the shop is referring to. If that is the problem, your client has two choices. She can knot and bury them (best) or put a dot of fray check, let it dry and then snip but I am loath to recommend that method as the fray check is a permanent glue and does leave a hard spot. I, personally would never resort to fray check to lock threads.
I think the quilt shop has needlessly caused your customer angst but now the seed is sewn and you have a very unhappy client. Even if the quilt is structurally sound, she no longer thinks it is and the shop has successfully convinced her that the quilt is "ruined". Far from it from what I can tell but you need to make things right. I would ask for the contact info at the quilt shop and contact them to ask specifically what "major issues" there are. I think you would have noticed if seams were popping open when you quilted it so if that isn't the case you need to make your customer happy. You can offer to pay for shipping back and forth and fix the issue, you can refund your customer or give her a substantial discount but I don't think that will make her happy because now she thinks the quilt has "major issues" that will cause it to fall apart. You are really stuck between a rock and a hard place. It is a hard lesson when a LAQ has a dissatisfied customer and sometimes no matter what you do they will probably never give you another chance.