One of the Images in the Description on the Mod Portal say not to use the default Menu.
This Menu can be toggled for any entity of type "linked-container" by setting at data stage the parameter "gui_mode" to "none", the default is "admins", optionally there is also "all" if someone would want to make it available to every user in a multiplayer game.
However if you want people not to use it then setting it to "none" so its not there in the first place would probably be a good idea.

Yes thats correct, however if you set the "gui_mode" to "none" (so the original gui for the configuration isnt toggled) you cannt paste blueprints with the my chests. So if you paste a chest with id 15 over a chest with id 0 the id isnt changed.

Edit:
Just tested to confirm: If the "gui_mode" is set to "none" the "link_id" of the copied chest is not set on the pasted chest. This happens with copy paste (SHIFT + Right Click/ SHIFT + Left Click) and when pasting blueprints new or over existing chests.

thats true, since people wouldnt be given access to it obviously they cant change the id either.
maybe you could do it via scripting in the mod tho that you listen to people performing the action and manually update the info that the game didnt update.
optionally i guess you could test if you can set the GUi to invisible while its enabled, which would let people update the link id in the described example but still stop people from usign the wrong window.
i have only briefly messed with linked chests as i attempted to do a similar thing but failed, so im not 100% confident if my ideas work or not.

I havent found a way to edit the "gui_mode" with scripts given that you have an entity when the event is triggered. And the LuaEntity class doesnt have the "gui_mode" variable.