You should state if the specification context for the work is AWS or ASME or what else. You should check if there is any contract language or requirements or local jurisdictions that supports what the inspector says. You can ask the inspector to show you the documentation that supports his interpretation.

In general, I interpret that the signatory for certification need only be someone who is authorized by the Company which "owns" the procedure or performance qualification. I am an engineer, I am not an "inspector" in any professional sense, but my signature certifies the accuracy of these documents (ASME basis) for our company. I have seen other welding procedure and performance qualification documents (context USA) that were signature certified by shop supervisors, QA managers, etc.