Unless other NDT methods are specified by the Engineer, VT is the only examination required unless the conditions of clause 2.19 come into play. The weld profile requirements apply, as does table 6.1.
As the inspector, your responsibility is to ensure the requirements of the drawing are met. If the Engineer or the designer do not do their job with regards to the joint details, there is little the inspector can do other than bring it to the Engineer's attention. The inspector does not work in a vacuum. There should be open lines of communication between engineering, design, and quality control.
Ultimately, the inspector is responsible for the inspections required by the code. The results and concerns are reported back to the Engineer. The Engineer has the responsibility and obligation to respond to the issues and nonconformances included in the report. It is the Engineer that makes the determination whether to rework, repair, or scrap the work. The method of repair is usually the contractor's responsibility.
Best regards - Al