You could remove the damaged border/batting right at the correct seam line plus seam allowance, then, looking at the quilt from the back, you could piece in a new section of batting (for each border around the quilt) and add the current border back on.
1. Looking at the back of the quilt: fold down the backing so it is not caught in the seam you are about to make. I would pin it.
2. Turn the quilt over and line up the batting section and the border right-side-together with the raw edge of the quilt where the original border/batting was removed.
3. Sew the layers together along the correct seam line all the way across the quilt. Open just to check that you have the border now facing up just like the rest of the quilt, with the batting behind it.
4. Repeat on all sides, trimming the previous batting/border even with the raw edge of the adjoining side before adding the new batting and border
5. After all sides have new batting and replaced border, you can reapply the folded over backing and re-quilt that area. It will look like nothing has ever happened to it.
Caryl Bryer Fallert used to employ this technique on most of her quilt iN the 1990s; don't know if she still does, but her quilts sold/sell in the 4 and 5 figure range!
http://www.bryerpatch.com/gallery/gallery.htm
Jan in VA