Thank you for your forum post.
I suspect that this is either a security permissions problem or a network / DNS problem.
The edition of SQL Data Compare does not change if you use it on the local machine or connect to it remotely.
I quote from your post:
It is only when I access it over the UNC path from another host that it is showing a different, standard edition key.
When you use the unc path connection, are the details of the standard edition license key identical to the local machine and view the professional license via the GUI ->Help ->about? If the license information is not identical, I suspect that DNS has not resolved to the correct machine.
If you ping the new Windows Server 2012 R2, Data Compare 10.7 host, does it resolve the correct machine name to IP address?