Building on Phil's answer:

Made the following change to the switch:

case 'extranet_sort_document_case':
add_filter('posts_join', 'extranet_clientdocument_case_join');
add_filter('posts_fields', 'extranet_clientdocument_case_fields');
add_filter('posts_orderby', 'extranet_clientdocument_case_order');
break;  

and wrote the following functions:

function extranet_clientdocument_case_join($join) {
    $join .= "LEFT JOIN wp_postmeta AS case_ids ON (wp_posts.ID = case_ids.post_id AND case_ids.meta_key = 'extranet_client_area') LEFT JOIN wp_posts AS case_names ON (case_names.ID = case_ids.meta_value) ";
    return $join;           
}

function extranet_clientdocument_case_fields($fields) {
    $fields.=", case_names.post_title";
    return $fields;
}

function extranet_clientdocument_case_order($order_by) {
    if(isset($_GET['order'])) $direction = $_GET['order'];
    else $direction = 'ASC';
    $order_by = 'case_names.post_title '.$direction;
    return($order_by);  
}

This worked in terms of making the column sortable by the post name of the related custom post, rather than by the metadata itself. However, it created a new problem in that the Title of the main post being listed, which is unusually the second column, now turned into the post title of the related post. I fixed this with the following additional code:

function restore_original_title($title) {
    global $wpdb;
    $post_id = get_the_ID();
    $get_document_name = $wpdb->get_results('SELECT * FROM `wp_posts` WHERE `ID` = '.$post_id);
    $title = $get_document_name[0]->post_title;
    return $title;          
}

and added the following additional filter:

add_filter( 'the_title', 'restore_original_title', 10, 2 );

Building on Phil's answer:

Made the following change to the switch:

case 'extranet_sort_document_case':
add_filter('posts_join', 'extranet_clientdocument_case_join');
add_filter('posts_fields', 'extranet_clientdocument_case_fields');
add_filter('posts_orderby', 'extranet_clientdocument_case_order');
break;  

and wrote the following functions:

function extranet_clientdocument_case_join($join) {
    global $wpdb;
    $join .= " LEFT JOIN ".$wpdb->postmeta." AS case_ids ON (".$wpdb->posts.".ID = case_ids.post_id AND case_ids.meta_key = 'extranet_client_area') 
               LEFT JOIN ".$wpdb->posts." AS case_names ON (case_names.ID = case_ids.meta_value) ";
    return $join;           
}

function extranet_clientdocument_case_fields($fields) {
    $fields.=", case_names.post_title";
    return $fields;
}

function extranet_clientdocument_case_order($order_by) {
    if(isset($_GET['order'])) $direction = $_GET['order'];
    else $direction = 'ASC';
    $order_by = 'case_names.post_title '.$direction;
    return($order_by);  
}

This worked in terms of making the column sortable by the post name of the related custom post, rather than by the metadata itself. However, it created a new problem in that the Title of the main post being listed, which is unusually the second column, now turned into the post title of the related post. I fixed this with the following additional code:

function restore_original_title($title) {
    global $wpdb;
    $post_id = get_the_ID();
    $get_document_name = $wpdb->get_results('SELECT * FROM `wp_posts` WHERE `ID` = '.$post_id);
    $title = $get_document_name[0]->post_title;
    return $title;          
}

and added the following additional filter:

add_filter( 'the_title', 'restore_original_title', 10, 2 );

Building on Phil's answer:

Made the following change to the switch:

case 'extranet_sort_document_case':
add_filter('posts_join', 'extranet_clientdocument_case_join');
add_filter('posts_fields', 'extranet_clientdocument_case_fields');
add_filter('posts_orderby', 'extranet_clientdocument_case_order');
break;  

and wrote the following functions:

function extranet_clientdocument_case_join($join) {
    $join .= "LEFT JOIN wp_postmeta AS case_ids ON (wp_posts.ID = case_ids.post_id AND case_ids.meta_key = 'extranet_client_area') LEFT JOIN wp_posts AS case_names ON (case_names.ID = case_ids.meta_value) ";
    return $join;           
}

function extranet_clientdocument_case_fields($fields) {
    $fields.=", case_names.post_title";
    return $fields;
}

function extranet_clientdocument_case_order($order_by) {
    if(isset($_GET['order'])) $direction = $_GET['order'];
    else $direction = 'ASC';
    $order_by = 'case_names.post_title '.$direction;
    return($order_by);  
}

Building on Phil's answer:

Made the following change to the switch:

case 'extranet_sort_document_case':
add_filter('posts_join', 'extranet_clientdocument_case_join');
add_filter('posts_fields', 'extranet_clientdocument_case_fields');
add_filter('posts_orderby', 'extranet_clientdocument_case_order');
break;  

and wrote the following functions:

function extranet_clientdocument_case_join($join) {
    $join .= "LEFT JOIN wp_postmeta AS case_ids ON (wp_posts.ID = case_ids.post_id AND case_ids.meta_key = 'extranet_client_area') LEFT JOIN wp_posts AS case_names ON (case_names.ID = case_ids.meta_value) ";
    return $join;           
}

function extranet_clientdocument_case_fields($fields) {
    $fields.=", case_names.post_title";
    return $fields;
}

function extranet_clientdocument_case_order($order_by) {
    if(isset($_GET['order'])) $direction = $_GET['order'];
    else $direction = 'ASC';
    $order_by = 'case_names.post_title '.$direction;
    return($order_by);  
}

This worked in terms of making the column sortable by the post name of the related custom post, rather than by the metadata itself. However, it created a new problem in that the Title of the main post being listed, which is unusually the second column, now turned into the post title of the related post. I fixed this with the following additional code:

function restore_original_title($title) {
    global $wpdb;
    $post_id = get_the_ID();
    $get_document_name = $wpdb->get_results('SELECT * FROM `wp_posts` WHERE `ID` = '.$post_id);
    $title = $get_document_name[0]->post_title;
    return $title;          
}

and added the following additional filter:

add_filter( 'the_title', 'restore_original_title', 10, 2 );