I've received feedback from our Second Tier Support team. They have noted that this issue should be resolved once WPML 4.7 is released.
Specifically, the issue occurs when using the Edit Translation link on the front end. While translation edits are saved, they are not delivered to the site until the back end is visited and refreshed. Additionally, the Advanced Translation Editor (ATE) fails to redirect correctly during this process.
A preferable alternative would be for the client to access the translation for editing from a location other than the front-end Edit Translation link.
Meanwhile, here are the recommended steps to follow until the issue is fixed:
1. Go to hidden link
2. Click "Edit with Elementor"
3. Add the desired text
4. Navigate to the French version at hidden link using the backend editor
5. Click on the English Translation in the sidebar to activate the Advanced Translation Editor
6. Correct the translation accordingly
Please confirm if this resolves the issue for you.