How to manually trigger a redirect to another page without changing the url address?

You can try to use the template_redirect hook to check your conditions and then set the wp_query->is_404 variable, like below:

function wpse202198_set_404(){
      global $post;
      global $wp_query;
      if( $post->ID == PAGE_ID_OF_PROTECTED_PAGE ){
           $wp_query->is_404 = true;
           return;
      }
}
add_action( 'template_redirect', 'wpse202198_set_404' );