Translate customizer settings with the WPML plugin
Make sure to have the WPML String Translations plugin installed.
1. Go to WPML -> String Translations
2. Click the "Translate texts in admin screens" link at the bottom of the page
3. Scroll to the "+ theme_mods_NAME" link and click on it (where NAME is the theme name)
4. Check the checkboxes of the "featured_button_text" and "featured_button_url" settings
5. Scroll to the bottom of the page and click the "Apply" button and wait till you see the success notice.
("featured_button_text" and "featured_button_url" can be replaced with any other setting you want to translate from the customizer)
6. Go to WPML -> String Translations again
7. You'll find 2 new rows in the "admin_texts_theme_mods_NAME" table (where NAME is the theme name).
If you don't see them in the list click on the "All Domains" dropdown field and select the settings from there.
8. Click on the translation link of each new setting and the field to translate will appear.