SQL Managed Instance currently doesn't support aliasing, so you will need to connect to the instance using the host name provisioned in the portal.
For Database Mail can always configure the IP address directly. But do you have a Domain Controller VM with DNS configured as the DNS server for your VLan?
Thank you for the reply. No, I do not have a Domain Controller VM. However, I have moved beyond the need for that; I got the connection string working with the unaliased MI instance name. And I removed my OnPrem DNS server from the VLan for both my web server
VM and for SQL MI. I was trying to use the VM as mail server for DB Mail. But it is still failing, giving the public IP address of the VM instead of the private IP. Do I need to restart the MI instance in order for the DNS server change to take effect? I don't
see any way to do that through the portal.