We show how to "rename" a disk on a running VM here.
It actually involves copying the disk and recreating the VM.
The details of the approach are in the following blog post.
The initial deployment of the VM, its disks, its NIC, VNET, etc. is here:
The script to delete the VM is here.
The deployment of disk copy & VM re-creation is here:
The script to delete the old disks is here.