I use "Move" on a near weekly basis on a Server 2012 R2 Datacenter standalone server. I have 2 tiers of storage and often use the "Move" command to relocate a VM from a fast tier to slow or vice-versa depending on what that OS is doing at the time. While this command can ALSO be used in "live migrations" between servers (if you've enabled the checkbox in settings to allow live migrations), it's normal purpose is to move a virtual machine to another directory or drive within the same server.
In a fresh install of 2012 R2 Std installed this morning, I see "Connect", "Settings", "Turn Off", "Shutdown", "Save", "Pause", "Reset", "Checkpoint", "Move", "Export", "Rename" in the list of options when clicking on a newly provisioned VM.
I'd be incredibly surprised if this option was missing from your list Fritz as I've deployed hundreds of Hyper-V servers and all have this option.
Regardless, it seems that the Export/Import has worked for you but, as you've found, the issue is that while importing you cannot overwrite an existing VM w/o:
A) deleting the original vm.
B) Importing using a new ID which also generates a new MAC and, if this is a domain controller, can royally mess with things as this is detected by Windows as a new clone.
C) Running three separate commands to: export, delete, import