I would design a quilt with a foundation peicing somewhat like a crazy quilt would be done. Foundattion would be a good qaulity muslin prewashed and ironed. I would use a fine needle in the machine and it should be fine but would test a small area first. I would not try to repair the damaged tape. I would work around the damage. This quilt when finished should be displayed and not used I don't think it will hold up with wear.
Glenn