Nice. I cannot close this ticket because of a Javascript error in the reply form. So i will just post my answer here. You may close this ticket after you read it.
Unfortunately, WPML does not support native WP features such as a different archive slug than the post type slug.
This is really an issue since many would want "/books" for the archive page, and "/book/harry-potter/" for the single post. With WPML you have to choose, because it relies on the rewrite=>["slug" => __('book')] value.
I hope this will be fixed, since it is a native WP feature.
If i use "has_archive" = __('books') without WPML, and switch the WP language, it will work as intended and and the archive will be named and translated correctly.
