It's a beautiful yo-yo quilt.
I don't think a sheet would be any heavier than fabric that is sewed together to make it wide enough for a backing.
Whichever you choose, for me, I would tack each and every yo-yo down. My machine has a tack stitch. A blending color of thread like off white, beige, light gray should work fine. Since each is attached to the other, how about tacking them right in the center hole area.
After all that, how are you going to bind the quilt?