Ok, I'm going to mark this as answered but, I believe the issue lies in user error 🙂

Original Method I was using to modify SQL Server Network Name:

1. Failover Cluster Manager (Win 2012) -> SQL Server Role -> click resources tab -> Bring Server Name Resource offline.

2. Right Click SQL Server Resource and select Properties

3. On the Properties tab modify VirtualServerName value

4. Click OK and receive error specified in question above.

Correct Way:

1. Failover Cluster Manager (Win 2012) -> SQL Server Role -> click resources tab -> Bring Server Name Resource offline.

2. Right Click on Server Name Resource for SQL Server Network Name and select Properties

3. On the General tab modify the DNS Name to the desired new name for the SQL Server Virtual Network Name

After bringing all the resources back online the virtual network name is successfully changed and accessible. Additionally, the @@SERVERNAME and entry in sys.servers is updated to the new name as well.