Pass custom css class to add_menu_page

You should not do this via CSS as the underlying functionality would still be there (and thus open for direct access).

I’d rather rewrite the theme in a way that the menu pages are only added if the current user has a certain capability.

if (current_user_can( SOME_CAP )) {
    add menu pages ...
}

You could even provide a new capability for your theme. Have a look over here, if you’re interested in this.