I am no expert, but the idea of using fusible on something so fragile could be fraught with difficulty. Would it be possible to apply a layer of organza (or similar fine fabric) over the top and applique it over the top of the damaged bits. It would be fiddly but wouldn't detract too much from the original.
Maybe if you had a spare room the quilt could go on a bed there...it sounds as if it is quite precious to you and it may last a little longer if not in constant use.