The tedious task of removing the entire border and sewing it back on again with the correct side facing up seemed to by my only option.
But then, I realized I had another option afterall.
I removed just the offending part of the border from the rest of the quilt.
And re-sewed the seam with the seam allowances on the other side this time.
Then I just re-attached the border to the quilt and no one will be the wiser. (except of course for you, dear readers, but you won't snitch, right?)
I'll post a photo of the full quilt top when I can spread it out and get a good shot. Until then, I just needed to share this experience with people who I know would appreciate it.