Quick update on anyone with the same issue:
I had a call with two CloudKit engineers last week and they basically confirmed my theory that something is corrupted. Essentially one of my homes is shared with an inexistant Apple ID (no clue how that happened, the other Apple ID still exists), so that's why iCloud servers are continuously responding with Not Authenticated and Internal Server Error.
In other words, there's nothing you can do client-side, you need to have a CloudKit engineer manually delete the broken home from your iCloud.
@Marco and @John, could I have a quick update on the deletion of the corrupted home? It's been a week and I don't have any other way to contact you other than this forum post. 😕