That was the solution.
I called Swisscom support today at 11 p.m. (I thought I finally had a reason to find out whether I was getting something for the Swisscom Premium, namely 24/7 support) and to my surprise, someone even picked up. The good woman wasn’t a professional, but she wasn’t nearly as overwhelmed as usual (in my experience) by everyone, regardless of the provider. She figured out what it was about pretty quickly.
She said so dryly: “This is all completely outside my area of expertise, but I’ll try to find out where I can screw things around.”
Without further ado, I told her the MAC from the Zyxel. She manually overwrote the incorrect MAC entered and after a restart, TADAAAA.
“O User Unser, holy be your word.” In the truest sense, thank you 😄