The backup fails with a Microsoft VSS Error, "A problem with Windows Shadow Copy was detected":
As the error suggests, if you take a look at the Hyper-V VMMS event log (Event Viewer → Application and Services Logs → Microsoft → Windows → Hyper-V-VMMS → Admin), you'll find the following event:
Source: Hyper-V VMMS | Event ID:
"Failed to mount the virtual hard disk '\\?\GLOBALROOT\Device\HarddiskVolumeShadowCopy885\monitoring server\Windows Server 2008 R2 Enterprise Evaluation (Full Edition).vhd' for virtual machine 'monitoring server': %4 (%5). (Virtual machine ID %2)."
This issue was caused by an issue from Microsoft, which was resolved by Windows Updates and patches.
In order to resolve this error, ensure that you install all Windows Updates including Optional ones. In addition to that you can take a look at the following TechNet thread showing 3 hotfixes that should be applied as well.