Passing a query to a module_hook that uses a second database

This is not an answer, strictly speaking, but rather a suggestion.

The thing is that your hook receives a $query already built with the original database connection, this explains the errors. To be effective, db_set_active() should be called before calling the Query constructor.

Otherwise, you would have to override constructors for both the Select class and its parent class Query, or more precisely "reconstruct" the Query builder instance for Select statements (and probably others too if you need range query etc.).

Also, unless having explicit requirements to provide a dedicated hook in this situation, it is not necessary.

For example it would be simpler to do something like this :

connector.module :

/**
 * Implements hook_init()
 * 
 * Adds secondary database connection information. 
 */
function connector_init() {
  $database_info = array(
      'database' => variable_get('connector_db_name'),
      'username' => variable_get('connector_db_user'),
      'password' => variable_get('connector_db_pwd'),
      'host' => variable_get('connector_db_host'),
      'driver' => 'mysql',
  );

  Database::addConnectionInfo('secondary_db', 'default', $database_info);  
}

/**
 * Switches from primary to secondary database and the other way round. 
 */
function connector_switch_db($db_key = 'secondary_db') {
  return db_set_active($db_key);
}

Example :

// Set secondary db active. 
$primary_db_key = connector_switch_db();

$query = db_select('registration')
        ->fields('registration')
        ->condition('id', 46, '=');

$result = $query->execute()->fetchAssoc();

// Switch back to the primary database. 
connector_switch_db($primary_db_key);

This softer approach also prevents hardcode like (...)->execute()->fetchAssoc() and let the implementation free to execute and fetch the results as needed.

For future reference, this is what I came up with (caching bits included):

function connector_init() {
    foreach([
        'connector_db_host',
        'connector_db_name',
        'connector_db_user',
        'connector_db_pwd',
    ] as $var) {
        drupal_static($var, variable_get($var, ''));
    }

    Database::addConnectionInfo('secondary_db', 'default', [
        'host' => drupal_static('connector_db_host'),
        'database' => drupal_static('connector_db_name'),
        'username' => drupal_static('connector_db_user'),
        'password' => drupal_static('connector_db_pwd'),
        'driver' => 'mysql',
    ]);
}

function connector_api_db($query) {

    $result = [];

    $sql_query = $query->__toString();
    $arguments = $query->getArguments();

    if (count($arguments) > 0) {
        ksort($arguments);
    }
    $cache_id = 'sql_' . md5($sql_query . '|' . print_r($arguments, 1));

    $cache = cache_get($cache_id, 'cache_connector');

    if ($cache && $cache->expire >= time()) {
        $result = $cache->data;
    } else {

        db_set_active('secondary_db'); // Switch to secondary db */
        $query_failed = FALSE;

        try {
            $db_result = db_query($sql_query, $arguments);
            $db_result = (0 === $db_result->rowCount()) ? [] : $db_result->fetchAssoc();
        } catch (\PDOException $e) {
            $db_result = [];
            $query_failed = TRUE;
        } finally {
            db_set_active(); /* switch back to default */
        }

        $result = (object)[
            'query' => $sql_query,
            'arguments' => $arguments,
            'result' => (object)$db_result
        ];

        if (!$query_failed) {
            cache_set($cache_id, $result, 'cache_connector', strtotime('+10 minutes'));
        }

    }

    return $result;

}