Replace the updateSite() in \Blt\Plugin\Commands ; Line 22 will ensure the drupal cache is cleared for the respective multisite.

updateSite

/**
   * Execute updates on a specific site.
   *
   * @param string $multisite
   *   Multisite.
   *
   * @throws \Acquia\Blt\Robo\Exceptions\BltException
   */
  protected function updateSite($multisite) {
    // Workaround to perform db updates before site context.
    $this->say("Performing DB Updates on <comment>$multisite</comment>...");
    $root = $this->getConfigValue('repo.root');
    $this->taskDrush()->dir($root . '/docroot/sites/' . $multisite)
      ->stopOnFail()->uri($multisite)
      ->drush('updb')->run();
    $this->switchSiteContext($multisite);

    if ($this->getInspector()->isDrupalInstalled()) {
      $this->say("Deploying updates to <comment>$multisite</comment>...");
      $this->invokeCommand('drupal:update');
      $this->say("Finished deploying updates to $multisite.");
      $multisite_cache = $this->taskDrush()->dir($root . '/docroot/sites/' . $multisite)
        ->stopOnFail()->uri($multisite)
        ->drush('cr')->run();
      if ($multisite_cache->wasSuccessful()) {
        $this->say("Drupal cache cleared for $multisite.");
      }
    }
    else {
      $this->logger->warning("Drupal is not installed for $multisite. Skipping updates.");
    }
  }