I have successfully set up iCloud custom domains with a different domain, so I do know how to do this and it's not an issue with my registrar or DNS records.
For one domain, I can do everything except add "John @ bergmayer.net" (spaces added just for here)--which is my actual primary personal email.
First, I got an error and couldn't even add it. Now, I can add it, but can't verify the address.
I thought it was because that address is already a contact method on my account, but removing that did not change anything.
I suspect it may be because that email address used to be my login for iCloud, before I switched my iCloud username to just be my iCloud email address.
*(This exact issue has caused problems for me before--for example, it hasn't been possible for me to subscribe to Hulu via in-app purchase because I once did, then cancelled, under the previous username, which seems to ***** whatever checks for a previous free trial or something. Naturally, both Hulu and Apple blame each other, which is depressing for the prospects of this new issue actually being resolved.)