We do not send subcontract CoC's to the customer, although we may pass through material certs and lab reports. When the part is completed it is our part, therefore we only provide a coc from us. We have had customers find out who our subcontractors were and cut us out of the loop by making some of the part themselves and then hiring our subcontractor.
I have gotten some heat at times from our customers but in this area management is very supportive and we have never had to provide that information since.