What is the best way to move MoonBot from one dedicated server to another?

You should close the MoonBot desktop terminal.
Then make a copy of the MoonBot terminal folder and delete the entire logs folder from it, it can weigh a lot and moving the copy to another server can take a long time.
To make the MoonBot folder even lighter, you can delete files with charts like: “BTC-MarketData.zip”, then compress the folder with the bot: right-click on the folder, select “Send” -> “Compressed ZIP folder” in the menu, then move this ZIP folder to a new server and uncompress it into a normal folder.
You can then run MoonBot.exe in administrator mode on the new server.