After resetting WPML, I was able to proceed with setting up WPML on your site's copy on my local server. I'm consulting our second-tier supporter about this case and will update you here once I have their reply.
Our second-tier supporter noticed the following message in the backend.
WPML is missing some records in the languages tables and it cannot fully work until this issue is fixed.
Please go to the Troubleshooting page and click on "Clear language information and repopulate languages" to fix this problem.
This warning will disappear once this issue is fixed.
Please see the attached screenshot wpml-is-missing-records.jpg.
But when he tried the 'Clean up' option in WPML -> Support -> Troubleshooting, it didn't help to solve the problem. As we mentioned with the copy of your site on our local server, we could fix this issue by resetting WPML. Our second-tier supporter suspects that the user has no access to write data to the database.
Could you please find out about it with the support of your site's hosting service?
We can also try to get more information about this by checking for any PHP errors while using the 'Clean up' option. Here are the instructions.
Please share with us WordPress's debug log (not WPML debug information). Please check this page for instructions: https://wpml.org/documentation/support/debugging-wpml/.
To enable it, open your wp-config.php file and look for
define('WP_DEBUG', false);
Change it to:
define('WP_DEBUG', true);
// Enable Debug logging to the /wp-content/debug.log file
define('WP_DEBUG_LOG', true);
// Disable display of errors and warnings
define('WP_DEBUG_DISPLAY', false);
@ini_set('display_errors', 0);
After this, repeat the action that causes the problem on your site. Go to WPML -> Support -> Troubleshooting, and click the 'Clear language information and repopulate languages' button. Please see the attached screenshot, clean-up.jpg. The errors will be saved to a debug.log log file inside the /wp-content/ directory. Then, please share the errors with us.