The relevant clause of AS9100 Rev D that discusses this is Clause 8.6, Release of Products and Services. In this clause there is a discussion on not releasing the product or service until all planned arrangements are met, or any difference is agreed to, and ends with the requirements to have documented information on the release of the product or service, which is often a C of C in the aerospace industry as you have indicated (even though AS9100 does not use this term). The requirements for this documented information is that it includes a) evidence that the product and service is conforming, and b) traceability to the person(s) authorizing release.
So, AS9100 does not say who that authorizing person needs to be, it is up to you to determine this. In general, what you would need to do is determine the competence that is needed for authorizing the release, and then dete rmining who has that competence. If you can show that the documentation tech has the competence to look at all of the documented evidence that shows that the machined parts are conforming to all requirements and determine that what you are sending meets those requirements or has been accepted otherwise, then you will have an argument that this person can do this task. Of course, this does always link back to the customer contract, so if they identify who needs to sign the C of C then there is nothing to really justify. If they ask for the quality representative, then it is hard to argue.
If you want more information on the quality representative in the AS9100 QMS take a look at this article: https://advisera.com/9100academy/blog/2017/09/25/who-is-the-best-person-to-be-as9100d-quality-representative/