Evidently the Level III held ASNT certification at one time. Whether he is still legitimately a Level III for his employer depends on several other considerations.
In your case, it all depends on the specifics of the contract.
If the governing documents invoked by the project specification or code is SNT-TC-1A, the employer can develop a written practice that permits the employer to define the qualification requirements and it allows the individual to be certified by the employer. However, your customer has the right and the responsibility to review the written practice and accept or reject it based on whether it meets their needs and expectations.
Many project specifications make the recommendations of SNT-TC-1A mandatory requirements, in which case, SNT-TC-1A is no longer a recommended practice, it becomes the minimum requirements that must be met.
Best regards - Al