What does your Purchasing Procedure say? Does it state different criteria for purchased parts vs supplied parts? If not, then either the customer must compy or the procedure needs to be updated to include what you need to do for customer supplied parts.
As a CM, we also receive parts from the customer. On the item card in ERP, we list this as a "customer supplied part" and they take responsibility for the initial product quality. We inspect for shipping damage and count. State on the RI record the items that are missing, like C of C, correct packaging and the like. We do discuss this with the customer and understand what inspection they perform on the part before they deliver it to us. Everything is clearly stated in the purchasing procedure, purchasing agreement, and understood by both parties. If traceability is needed, we assign an in-house number that can be traced back to the PO, which states that no C of C was delivered and traceability to the original manufacturer is the responsibility of the supplier. For supplier quality, the customer gets the same report card as the rest of our suppliers. Bad product, poor OTD, it all gets reported.