Thank you for the details, I did not get notified about the post update for some reason.
Okay, so I've made an archive.php file in your child theme directory, where I placed a small custom HTML and CSS snippet containing the search bar, similarly positioned and styled as on the other pages. Now you should see the search bar on the archive pages as well.
Please note that since this qualifies as a customization request, which is not part of the support process, I can't guarantee that is solution is the best or if it will work for all cases.
I hope this helps!Best,
If you like my products, don't forget to rate them on codecanyon :)