ahhh, that is so cute!
I liked bearisgray's idea of adding patches or bias tape ... if the fabric is as bad as you think, you might consider insuring the patch is glued in those frayed spaces and then "applique" along the bias strip or patch.
My mother bought a quilt from a church group for the kids one year.
as your bear is suffering, so did this blanket. Taking a hint from "raw-edged" quilting, I have just started taking larger blocks, and putting them over the raggedy patches and stitching them down with a small zigzag stitch. I figure within three or four more years, I will have a new quilt, with the old one quilted inside of it, the places that I sewed the new blocks in will be the quilting to replace the tying that was done in the first place.
(As an aside - - I built my grandson a patriotic bed quilt back in 1996 from fabric I bought at WalMart. I used Wright's ribbon binding ... blanket binding I guess is the better word/name. As a 2-yr-old child growing up with that quilt on his bed, you can imagine the wear, tear, and washing that quilt has gone through! The blanket binding is absolutely THREADS! But, there isn't one seam or piece of fabric that has given way)
Do let us know what you finally decide to do with your little teddy.
Have you considered (this always happens when I spend a lot of time TALKING) ... have you considered sliding some fusible web under the places that are separating and fraying, using tweezers or nail file to smooth it out?
If you insert the fusible first, with the glue facing outward, place the patch in place, do your appliquing, and then apply the iron to stabilize the whole thing.