Found some working solution!
I just created child-templates with header.php only for each individual language and applied the hard-coded cookie write inside of them. Detail are in following article:
http://dallascao.com/en/use-cookies-to-remember/
Hi, I’ve tried everything and finally, there’s the solution of my problem.
I had to add the function
<?php wp_footer(); ?>
to the footer of my custom template.
Now it works as a charm