"Problem Turning On Mail" when creating an email address for iCloud Mail (for custom domain)

Note, my setup has the aforementioned gmail AppleID but ALSO has iCloud signed in to the other AppleID. The solution suggested by zinacef was unrelated as it's only true if the domain of the Apple ID is iCloud.

user7284's solution that can work is shown but painful. First, go into Mail and change the description for the iCloud account and see if that does it. If not, the simple solution is to create a new, temporary account on a Mac (borrow one if needed). Sign into your Apple ID and try it there. (Might even work on Windows with the Apple software installed.)

In my case, my setup has the aforementioned gmail AppleID but ALSO has iCloud signed in to my old AppleID for just iCloud mail. This new solution on the temp account didn't know about that account so it allowed me to turn on a new primary account with the new name. I reopened Mail and it appeared alongside my old one. I logged out of the temp account and reopened Mail on my main account and everything is now happy.