No, in this situation the operation would not be required to have the Pack ID on the boxes. The requirement in Section 17.2 (Use of Packaging Materials) requires a Pack ID on market ready packaging material under the assumption that the operation is not able to put it on the actual product. If this information is on the actual product, this is even better for traceability and fulfils the requirements.
As per Section 17.2, the box would still need to be labelled [unless the secondary container is transparent (e.g., a large clear plastic bag holding smaller labelled bags of carrots)] with the correct identifying information (i.e., name and address) of the operation that produced the product, OR the operation that packaged the product, OR the company for whom it was produced/packaged. This is a requirement as well as a federal regulation.