With it being such a tight fit I thick I would take a minute & add a 2-3" border around the backing. That way you have plenty of wiggle room. I've done this before on some small quilts where I had a special fabric but not quite enough. I pressed the seams open & then after quilting just trimmed the backing.
You could even just add a strip to 2 sides (1 in each direction) and then it would look like part of the design for the back.