So sorry about your grandmother.

I would hand quilt it with muslin as the backing and either flannel or a white cotton batting inside. I would lean more toward flannel inside to match the age of the quilt. Echo quilting would be good.

At first i was thinking binding by sewing scallops to flip over on the back, but this is essentially the same edge as you would get with birthing, which would give you the full back. The advantage of using a muslin binding (instead of birthing) would be to protect the edges from wear.

That is so beautiful and a treasure to be had.

Thank you all for your suggestions. I think mom will back it in muslin as that is what the front uses. We'll work together on the quilting so that will be fun.