You'd have to ask an attorney, but I believe that any service organization that accepts a product from you is responsible for the part/product until it is returned to you. Bottom line being that if it's damaged, be it in transit or in their hands. it's between the service shop and shipping company. If you've fully notified both parties I believe you can hold the service company accountable. As the shipper it's on him re: shipping service, collection from FEDX, etc. They may not try to say no, but I don't think they can.
Probably worth a call to a lawyer cause I could be wrong on this pretty easily.