Dear Niklas Rechter,
Good day! Thank you for posting to Microsoft Community. We are happy to assist you.
Based on your description, you are using accounting software (Billbee) to send the invoices directly to the customers automatically. And you are getting 554 error. (For your information security reason, I masked the private information in your screenshot to prevent phishing/spoofing risks, please comprehend.)
From the issue details, I was wondering do you have the SMTP server integrated with Microsoft365, as you want to contact Microsoft server? If yes, there is possibility that the sending IP/URL need to be delisted from the spam list of the server (backend side), please comprehend that our forum support has limited permission and could not access the backend server. In this situation, we suggest you contact the higher-level support for Office 365: Please go to Office365 admin center>Support>New Service Request. The support engineers there have the correct escalation channel, and this is the most efficient way to report such issue. You can refer to this article to check the way how to raise a ticket: Ways to contact support for business products - Admin help. The engineers in related team have higher permission and resources than us to help check the root cause from background directly.
If you are not using Microsoft365 or other Microsoft products, you may need to contact your server host for further support.
Hope the information can provide the best channel for you. If there is any update or I misunderstand you, please feel free to let me know. I will continue to assist you. Your understanding and co-operation are highly appreciated.
Thanks for your precious time. Have a nice day!
Sincerely,
Sherry | Microsoft Community Moderator