After two days, I actually found an error.
It was kind of hidden, because JSON.stringify() doesn’t include getter and their current values. Even iterating over the object via a for-loop didn’t show them. Checking on .constructor (.prototype doesn’t exist on this specific object) did the trick to find out its type and the corresponding getters.
The promise of fetch() rejects with a TypeError and the following message: “Failed to fetch”.
That seems to be a network error. But other extensions (and the Twitch app itself) doesn’t have any problems.