About the backing, you've received lots of good ideas ... ITA that the quilting would help determine the backing. You mentioned the stripe isn't as good of quality, so I'd be hesitant. Another factor to consider is getting the quilt squared up with the stripe.
You said you'd like a wider border but don't want to take off what you have.
Why not just add more borders?
Or add a simple flange in the white, and a wider strip of the red?