Hi fschuld,
Thanks for reaching out to Microsoft Community and posting your inquiry. We appreciate the effort in performing initial troubleshooting steps.
We also suggest you try a different outlet or a different Microsoft Surface Charger. Otherwise, this could be an issue with charger. Please contact the Microsoft Surface Support to check your options. If you're located outside the United States, the Global Support can also help.
Keep us updated about the issue by replying on this thread.
Kind regards,
John Neil_J