I would like to request temporary access (wp-admin and FTP) to your site to take a better look at the issue. It would be better to have a testing site where the issue is replicated. Your next reply is set to private to share the info.
❌ IMPORTANT: Please backup your database and website before proceeding ❌
You can use this plugin to create the backup: https://wordpress.org/plugins/duplicator/
✙ I would need your permission to de-activate and re-activate Plugins and the Theme to change configurations on the site if needed. This is also a reason the backup is critical.
The issue comes from the plugin Permalinks Customizer. I've disabled that plugin on your site and the language switcher works properly. Can you please confirm this?
This plugin is not available in our list of compatible plugins, which you can check here https://wpml.org/plugin/
So I suggest you contact the plugin author and ask them to join our Go Global program where our Compatibility developers will help them to make their product compatible with WPML. https://wpml.org/documentation/support/go-global-program/