With the server-side encryption, I’d go through the official webdav interface. You can try directly a client with webdav that can move files without downloading them (windows: winscp). Is this a fast connection between the server and the ftp storage?
I haven’t tried this recently, but copying a few files directly like this worked ok if the transfer speed is high (I used SFTP at the time and also a few GB of data). But speed between server and storage >100 Mbit/s. And I didn’t use the server-side encryption…