Custom menus are specific to a given layout so installing a custom men set on one layout has no effect on what menu set is in use on other layouts.
What you might do is have your script that currently installs this custom menu set on the first layout set a global variable to some specific value. Then, OnLayoutEnter triggered scripts can check the value of this global variable in order to detect these higher access users and install the different menu set.
Thanks for the tip that custom menus are specific to a given layout. I can take it from here.