Update for anyone who is having this same problem:
I still had the previous version, 1.0.4.37, on my hard drive, so I un-installed the new version, and re-installed the previous. Now the data copy works just fine. It’s a good idea to always keep the previous version on your local drive, since I can’t find a way to get the previous version on the WD site.
For now, when prompted to load the new version, just hit cancel.