Add custom taxonomy to post content body?

Even though extending your custom search would be the proper WP way to solve your problem, the answer to your workaround question would be the below function:

function change_content_on_save($post_id, $post, $update) {
  if ($_SERVER['REQUEST_METHOD'] !== 'POST' || empty($_POST))
    return;
  if (defined('DOING_AUTOSAVE') && DOING_AUTOSAVE)
    return;
  if (!current_user_can('edit_post', $post_id))
    return;
  
  // create your terms list and insert to content here:
  $post->post_content = $terms_list;
  
  // Delete hook to avoid endless loop
  remove_action('save_post', 'change_content_on_save', 10);
  wp_update_post($post);
}
add_action( 'save_post', 'change_content_on_save', 10, 3 );

save_post is an action triggered whenever a post or page is created or updated
https://developer.wordpress.org/reference/hooks/save_post/