Hi,
I used to run a web hosting company and we managed multiple servers with cPanel, Plesk and DirectAdmin software before the company was acquired by Serveriai.lt. Occasionally we had to upgrade our servers and move clients' websites from one cPanel powered server to another, causing little to know downtime. I did this countless of times already. I still do migrate our own websites. We did a transfer a 200 GB website we run just recently, causing no downtime at all. I'm confident I can help you migrate your data from server to server smoothly and without downtime.