CopyDatabaseWizard.exe issue on SQL 2014
As weird as it is, it currently appears I have resolved this issue by upgrading Powershell. On one server the powershell version was 3.0 and on the other three it was still 2.0.

You can check your powershell version installed by running:

$PSVersionTable.PSVersion


The upgrade package I used was Windows Management Framework 4.0, which includes Powershell 4.0. PS 4.0 is the last version supported for Server 2008R2. 
Here's a download link to WMF 4.0:
http://www.microsoft.com/en-us/download/details.aspx?id=40855

Edit

Credit to Antoine Hernandez; Powershell 5.0 is supported on Windows Server 2008r2.
Download: https://www.microsoft.com/en-us/download/details.aspx?id=50395

Also quick update after a few days: I haven't had any instances of this issue since I updated Powershell to v4.0.