I have a website in my company server and the customer recently moved to Azure.
The customer uses Visual Studio to track changes and any code must be registered in there.
They have prepared a LAMP Stack for our website using a private IP (so that it can only be accessed by users from inside the company).
The access is restricted a bit and we only get access via SFTP + VPN or through the web version of Visual Studio. There are two stages - Before it goes live we upload to staging and once everything is smooth in there, they copy all files to Production and it goes live.
I don't mind how you migrate the website (BackupBuddy, Duplicator, WP Clone, Manually), as long as it is up and running.
If it goes well, I have one more challenge like this one for you.