I think I found the solution... I had a Gmail account as my Apple ID but am using my iCloud account for my personal email and it was also the primary email address on my Apple ID. The Gmail account was solely used for the username of my Apple ID. I too was met with the error last night when trying to switch to my iCloud email address, saying something like "You cannot use an Apple ID ending in @icloud.com."
What I ended up doing was to go into Settings on my iPhone: Settings > [Your Name] > Name, Phone Numbers, Email and changed my primary email address away from the iCloud address I was trying to change my Apple ID to. Thinking this would solve it, I then tried the switch but was met with the same error message. I then went back into Settings on my iPad, MacBook Pro, iMac, and iPhone and checked out Messages and FaceTime. I found that on my computers, my iCloud email address was checked to send and receive Messages from. Odd, because I have always left only my phone number checked, but nonetheless I unchecked my iCloud email address. I then retried the change to my iCloud email on the Apple ID site but was met with the error. I then signed out and signed back in to the Apple ID site and was then finally able to change from the Gmail address to my iCloud address as my Apple ID.
Hope this helps!