Changing title bar in browser window of theme “Encounters Lite”

Filter wp_title, get the current page title from your external app, and return that title.

Sample code:

add_filter( 'wp_title', function( $title ) {

    // check if it the correct page
    if ( ! is_my_external_app_page() )
        return $title;

    return get_title_from_external_app();
});

The implementation details are up to you.