In order to hide the button menu, you need to override the basic functionality, which involves development within the system.
Firstly, you need to override basic schema LeftPanelTopMenuModule. Please find the loadMenu method. There you can see that visibility of the button in based on user type. You can add your custom method to check current user rights and set the visibility based on the response result:
Here is an example of how to check current user role. In the example we are hiding the Menu button form non-administrators role. You can adjust the code up to your needs. (the example involves creating of custom CSS style and adding it to the schema dependencies):
As for the list of menu items, please take a look at loadItemsMainMenu method, which is responsible for its logic.