Good evening Clemens and welcome to the Events Calendar Support forum!
Thank you for reaching out to us.
We are sorry to hear about the issues you have been experiencing while using our plugin with WPML. I would love to help you with this topic.
First off, I would highly recommend you read and apply the recommendations found here: https://theeventscalendar.com/knowledgebase/setting-up-the-events-calendar-with-wpml/
Secondly, at this point I would recommend switching to one the 2 other methods for multi-language support (directory or domain based). If nothing else, this is much better from an SEO perspective.
The other reason being that you are right, we currently do not offer full support for the ?lang= method.
Let me know if that helps.
Have a great day!
Geoff B.