It sounds like something on the server is blocking the request that saves menu settings. This can sometimes happen when the server is configured with a low request size limit or when a firewall incorrectly flags the request as suspicious.
Is this server running ModSecurity or something similar? If so, please check the relevant logs for recent blocked requests where the URL matches the AME settings page (look for wp-admin/options-general.php?page=menu_editor
). If you find any blocked requests like that, the log might also provide information about the rule or setting that caused the firewall to block the request.