First off, the GUI you posted is for Windows Server 2008. This server is 2012 and Disk Management is run through Server Manager on a different GUI.
The images posted are from a 2012 R2 VM.
In 2012 R2 Server Manager, click Tools\Computer Management to get to Computer Management GUI:
Second, there is no option to Shrink the drive in Disk Management. Only to Extend.
In 2012 R2 there is - see screen shot in my first post. If that option is not available in Server 2012, you can either upgrade to 2012 R2, or use 3rd party tools to either shrink the disk in the VM, or do an offline disk clone with resize.
Furthermore, when I try to Compact the .vhd in Hyper-V, it spits out an error. A rather nasty one which tends to crash all of Hyper-V.
Shrink and Compact are 2 different things. Compact is a process that applies to dynamically expanding disks only, and reclaims disk space occupied by the VHD(x) file that has been deleted inside the VHD(x). Compact must be done on offline disks, not ones
that are being actively used by Hyper-V. Compact is not what you want here.
Shrink is a process that reduces the size of a VHD(x) file. In a fixed VHD(x) file you'll see the size reduction on the Hyper-V host. In a dynamic VHD(x) file the actual file size will remain the same, but the size seen by the VM will show the new reduced
size. This is what you should be doing. If the guest OS (VM) has a partition on that VHDX that's taking all the space presented by the hyper-V host, the "Shrink" option won't be there. Hence, you must shrink in the guest OS first, then go out to
the host and shrink it there as well.
Here's an example of a Gen2 2012 R2 VM that I've shrunk its boot/system disk in the VM and on the host - with no down time:
The Shrink Process has nothing to do with how the disk is formatted in the guest OS. All it needs is free space (in 1 GB increments) on the VHDX file.