Here is the solution. I knew as soon as I asked the question I would find the answer!

First off, the query building code must not run pager_default_initialize, like so:

$total = $query->countQuery()
  ->execute()
  ->fetchField();

$page = pager_find_page();

$query->extend('PagerDefault')
  ->limit($per)
  ->range($page * $per, $per);

The pager render array should be created like so:

$content['results']['pager'] = array(
  '#pre_render' => array('CUSTOM_PRE_RENDER'),
  '#theme' => 'pager',
  '#variables' => array(
    'total' => $total,
    'per' => $per,
  ),
);

And then run pager_default_initialize in the custom pre render function:

function _document_library_intialize_pager($_element) {
  pager_default_initialize($_element['#variables']['total'], $_element['#variables']['per']);

  return $_element;
}

This code will only work to add a single pager to the block and may not work if there is more then one pager on the page.

Here is the solution. I knew as soon as I asked the question I would find the answer!

First off, the query building code must not run pager_default_initialize, like so:

$total = $query->countQuery()
  ->execute()
  ->fetchField();

$page = pager_find_page();

$query->extend('PagerDefault')
  ->limit($per)
  ->range($page * $per, $per);

The pager render array should be created like so:

$content['results']['pager'] = array(
  '#pre_render' => array('CUSTOM_PRE_RENDER'),
  '#theme' => 'pager',
  '#variables' => array(
    'total' => $total,
    'per' => $per,
  ),
);

And then run pager_default_initialize in the custom pre render function:

function custom_pre_render_intialize_pager($_element) {
  pager_default_initialize($_element['#variables']['total'], $_element['#variables']['per']);

  return $_element;
}

This code will only work to add a single pager to the block and may not work if there is more then one pager on the page.