WPML may change settings during plugin updates. If the below settings are not as in the screenshot, please confirm with WPML support how to translate the slug for custom post type. For sure it’s possible as it was working when WPML provided help instructions below.

About the permalinks, you can translate the permalinks for the custom posts following these steps:

– Go to WPML-> Translation Management -> Multilingual Content Setup tab

– Go to Custom posts slug translation options and enable that option

– When the page reloads, you can add a translation for the custom post slug (as shown in the screenshot).

wpml slugs

For the taxonomies permalinks, it’s impossible to do that (while you can translate the terms for sure).