venimo
You can look at @meditatingsurgeon rsync method, but I would be a little skeptical for enhance, as Adam has said it needs a blocklevel copy. There is a way of using restic or borgbackup to get a better blocklevel backup of a running server. Hetzner even have a guide for borgbackup to their free ssh storage.
@xyzulu is right with DD but that's way too complicated unless your cli foo is great.
I suggest using Clonezilla, it can be mounted from image usually, or worse case you can actually edit your grub file to manually add the iso as an option. From there just do a disk to image copy, once that's fully worked, you'lll be able to use Clonezilla to do the reverse. If you have issues with Clonezilla due to bad sectors, say you have a failing drive use DDRescue instead of Clonezilla. It's magic on foobar'd drives. Of course make sure the CP server is shutdown before bringing up the new cloned one.
If I remember correctly, this should be all you need, BUT once that's systems moved and up and running it will try contacting other systems in the estate. (from its. new IP) I'd be inclined to shut the estate down until you know you've brought up the cloned CP in a fully working state. Once happy bring the other servers up and let them get the new IP from the CP when it talks to them.
That's how I would do it, given you have IPMI/KVM access this should't be impossible for you. That said I know what I am doing and I am still paying for a server as I don't want to risk messing it up. I am just waiting for the CP migration/DR feature.
Maybe ask Adam for any extra information to help you.