It may look like a strange request from the customer of the customer but as other have said it is certainly driven by a CAPA at the end user which is not visible due to being twice removed.
I do not question that there are better ways then the one they requested. They are writing procedures to control a process that they are twice removed from, your process.
The process you were using of a high quality calibrated reference to verify the tooling gauge periodically makes plenty of sense and is easier to manage with less risk.
Still, the customer is always right
As long as they are paying for it...
Note, we do not say the customer is always wise