WPML and custom post types archive template
I found this support thread where they say that one should change the following line (in your code):
'has_archive' => 'case-studies',
to:
'has_archive' => icl_translate('wpml_custom', 'wpml_custom_showcases', 'case-studies'),
Might be a good idea to ask the official support for this, as it's commercial software and no documentation is available.
I found out what was the problem.
The string was not translated under WPML -> String translations
When I translated it (case-studies -> de/case-studies) it worked.
Actually, it worked in all variants - both my original code, and code suggested in the answers.
I think this will work for you, just missing some action :)
'has_archive' => 'case-studies',
to
'has_archive' => icl_translate('wpml_custom', 'wpml_custom_showcases', 'case-studies'),
THEN
Go to Settings > Permalinks and hit 'Save'.
Let me know if this works perfectly.
Cheers!