Cannot set up Cox.net email in Mail -- iCloud account conflict?

I'm pretty Mac-savvy but I'm having a heck of a time setting up Mail on a spanking brand new iMac for my brother-in-law today.

When I set up the iMac for the first time today, I chose the option to sign into his existing iCloud account. No problems.

Then, I wanted to set up his Cox.net email in Mail. And this is where I am stuck.

I could describe the multitude of variations of the problems I've encountered trying to set this up, but it seems to me that it all boils down to one thing: the email address for the iCloud account and his Cox.net email address are the same.

First, I tried adding a new account, as all the help pages I've seen tell me, but it won't let me because there is already an iCloud account with that email address. So then I figure that okay, it must want me to use that iCloud account for BOTH the iCloud and Mail. So, I went into the account and added the Cox incoming and outgoing mail server info.

BUT...around the top right of the Mail screen it says "Login Failed." When I click that on, it tells me that my iCloud password is incorrect (it's not talking about the Cox password, but iCloud). So I enter the iCloud password in where it tells me to log into icloud again, and I'm in. But then every time I try to do anything in Mail, or click the "Login Failed" wording in Mail, then it wants me to log in once again to iCloud. I never logged OUT of iCloud, but it keeps wanting me to long in.

It seems like everything I read on setting up email in Mail ignores the idea of already having an icloud account, and show you how to do it if there is no existing iCloud account.

I'm pulling my hair out. Any help?

P.S. This is an area on Macs that I am most unfamiliar because personally I use gmail and just access it from the browswer -- never tried to set up an email account for Mail, nor do I use Mail.