Most museums with the ability to store a quilt in proper conditions will not take one in this condition. And most small local museums don't have the ability to keep it in such a way as to prevent further damage, unfortunately.
The idea to recreate it is a perfect one! The reproduction fabrics would be easy to locate and there are good quality muslins now that will take the wear for the background.
Make a memory box/shadow box in frame with pieces of the old quilt, photos of your GGM, the new fabrics, and photos of you remaking it to hang in the room where it will be used.
Jan in VA