I would cut off the binding, and put a new backing on it. Instead of stitching (which would show on the front), I would tie the backing onto the quilt, and have the ties show on the back only. You can put the ties close together, to help the quilt for stability, and you'll have a good fix to help stabilize the fragile fabric already on the quilt. That, with the new backing, should hold it together well for the future. Put the bindings back on and you're good to go.

I think I understand what you are saying about not affecting the design to the front. Inserting a photo of the front, apologies if this comes through as big as the post #1.
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Sounds like you have some good suggestions here - hope it all turns out well for you. What a bummer!! A friend of mine had this issue with some "old" black fabric -- she had pieced a quilt with it and it started to come apart when it was being quilted at the long armer. Hers was not able to be salvaged though.

Does he by chance have a cat? It almost looks cat scratched. Also, I think you should tell us which manufacturer so we can avoid their fabrics! There is nothing wrong with stating the truth about a particular brand, even if it's bad.

Also, if the problem isn't cat scratching, I would definitely contact the manufacturer and tell them about what happened. They might attempt to compensate you in some way, with perhaps some new fabric.

Otherwise, everything the others said. You may not have to completely replace the backing, just sew in a "patch" where the shredding is occurring and then hand quilt it to secure it...or ask your long-armer to do it. You could even tie it to secure the patch. Use a fun floss to tie it or use buttons. Doing this would also eliminate the need to remove the binding.

Also, cute quilt!

I've refronted and rebacked a number of quilts, works fine. It is a lovely quilt and a lovely gesture.
And yes, don't depend on the dark colors for longevity.

Hmmmmmm..... cutting of the binding will slightly affect the dimensions of the light blue stripe. It will not be the same size as the ones in the middle. But it would only be removing about 1/4". You could just undo the hand sewn part. With the binding "open" you can sew over the previous stitching of the binding then turn it over and re-sew the hand sewn part. Hope that's understandable. :)

That is a very nice quilt. I think I would remove as much of the backing as possible - think removing the paper from a paper pieced block. The threads that are under the quilting stitching - I would leave alone.

Sorry this happened to you.

You don't need to cut off binding just undo the hand stitching on the back and lay new backing on top then machine exactly on old row os stitching then hand stitch.
Have you thought about tying the quilt and not quilt.

This is what I would do too. It's only the hand sewing of the binding that needs to be un-sewn. I too think I would leave the misbehaving back on, add another backing and tie it backwards. Good luck; so sorry this happened!