AWS D1.1 does require that NDT personnel be qualified in accordance with ASNT recommended practice SNT-TC-1A. You need written documentation of certification. UT personnel are also required to pass a separate examination based specifically on the UT requirements of the D1.1 Code. This examination is usually employer based, but I have seen some that were client based. (Sometimes a demonstration of ability is also required). We document this examination on the qualification form.
As far as NDT procedures used, they must conform to the requirements detailed in Section 6. There are a couple of references to the fact that the NDT personnel should be familiar with the limitations of particular test methods.
To answer your question, since your firm has written a procedure, yes I would think that your Level III would sign off on it, as this is not in the jurisdiction of a NDT Level II or a CWI. From a contract point of view, The AWS procedures described in Section 6, Part D stand on their own. Your firm may be able to state in writing that "NDT procedures used by Level II NDT inspectors conform with the requirements of AWS D1.1-2004", or wording to that effect.
~thirdeye~