The outside calibrator is the manufacturer of the test unit and they are certified to the 17025 lab standard. Their software actually produces page two and three of the certificate package, but page one, the actual cert, needs to be dressed up to Name the calibration lab and place the serials and cal data as I see it.
But other than that we should be OK, right?
Idk, but lets break it down.
You make a wrench.
You calibrate the wrench during assembly based on calibrated testing equipment which is done by a 17025 lab.
You issue a calibration certificate that the wrench is "good" based on your testing.
Your customers complain because your certificate doesn't show you are 17025 accredited, because you are not.
They don't like it because now they have to explain something to "the auditor."
Well in order to meet the equipment manufacturer "exception" your customer has to ensure you meet the 7.1.5.3.1 requirements. Basically 5:
a) adequacy of procedures
b) competency of personnel
c) testing of the product
d) capability to perform correctly and traceability
e) customer requirements
f) review of related records.
I don't know if naming the calibration lab "works" because they are not calibrating the wrench. You're showing traceability which is on part. It sounds to me like you will have to show the balance of the requirements as well.
You may need to sit with your customer and figure out what they need. My guess is they haven't really thought about it.