Your inspector is doing a good job. There are a couple things to consider with your self-locking helicoil insert installation.
First, if my memory is correct, the deformed thread locking feature of your helicoil is located at the middle of the insert length. It is good design practice to ensure that the installed mating screw extends a minimum of two thread pitches beyond the locking feature, which in your example using a 6-32 screw would be .063" min.
Second, to obtain the full design load capability from your helicoil, the insert's entire thread length must be engaged by the mating screw.
Third, you should check that the mating fastener has sufficient thread length to prevent the threads from "bottoming out" under the worst case dimensional/tolerance stack-up conditions expected. It can be hard to determine a screw that has just barely bottomed out on its threads from one that has been installed correctly, especially when using torque wrenching.
All of these things will be affected by any difference in axial location of the insert relative to the attachment interface resulting from the installation procedure described. These seemingly minor design details can cause major problems unless you pay attention to them.