Override contributed module output (inline links)

I've installed the Drupal 7 port of the Node to Word module for exporting nodes as word or txt documents. The module packs two links - "Save as Word" and "Save as Text" - into the $content['links'] object in the node. I'd like to remove these links and output them as a variable that I can print where I like in my theme.

I've tried:

 function mytheme_preprocess_node(&$variables) {
   unset($variables['content']['links']['node']['#links']['node_to_word_link']);
 }

using my theme name for "mytheme", but it doesn't work.

I also tried:

function mytheme_node_view_alter(&$build) {
  unset($build['content']['links']['node']['#links']['node_to_word_link']);
}

but this just proved to me that I have no idea how this hook works.

I believe the module is using the following code to set the links:

function node_to_word_node_view($node, $view_mode, $langcode) {
  $links = array();

  if (node_to_word_enabled($node->type)) {
    if (user_access('access node_to_word docs')) {
      $links['node_to_word_link'] = array(
        'title' => t('Save as Word'),
        'href' => "word/$node->nid",
        'attributes' => array('title' => t('Save this document as a Microsoft Word document.'))
  );

  $links['node_to_word_txt_link'] = array(
    'title' => t('Save as text'),
    'href' => "txt/$node->nid",
    'attributes' => array('title' => t('Save this document as a plain text document.'))
  );

  $node->content['links'][$node->type] = array(
    '#theme' => 'links__node__blog',
    '#links' => $links,
    '#attributes' => array('class' => array('links', 'inline')),
  );
    }
  }
}

I don't want to use a css display:none hack, I'm trying hard to learn the php logic for Drupal. Might be an uphill battle - I'm a PHP noob. Any help is deeply appreciated.