My second Apple ID has disappeared from Apple Mail on my MacBook. How do I restart it?

I have been working on this problem for several weeks. After upgrading my HDD to an SSD, my me.com Apple Mail account disappeared from my MacBook somehow and I cannot make it appear again. It works fine on my mobile devices and my iMac, just not here.

I have two Apple ID's: the original based on Gmail decades ago (it is not the one used for Mail) and a me.com email that I use for my iCloud.com email using the Mail app. This is the one that has disappeared.

My second iCloud Apple ID (the one using the me.com email) was recently re-verified - after a week of "help" from the Community that did not resolve the issue and a 6-day waiting period from Apple after that - but when I try to add the me.com address using the iCloud instructions, the system does not recognize it. I can access my iCloud email using a browser but I want to use the Mail Apple app.

How do I get the system to verify this account?

I use macOS Catalina 10.15.6 and Mail 13.4.