Are you referring to a glue-laminated beam, or a laminated veneer lumber beam? They are two different things.
If it's a laminated veneer lumber beam, holes are permitted in the middle third of the length of the beam and the middle third of the depth of the beam only, and the space between the holes must be two times the diameter of the largest hole.
This applies ONLY to uniformly loaded beams! If there are concentrated loads on the beam, then the inspector is correct in requiring engineering.
You deal with "these guys" on a regular basis by humbling yourself before them, doing their work for them, and by not fighting them but complying with their wishes. You will find that they are often correct.
Every manufacturer of LVL's makes available a chart of permitted hole locations and sizes. It is easy to obtain, and will probably satisfy the inspector if you get one for him.
Architect (NY) and Home Designer (PA)