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I solved the problem, at least temporarily, as follows:

Navigate to the \Temp directory specified in the error message (C:\Users\username\AppData\Local\Temp\).
Look inside that directory for a subdirectory called "BaseCamp."
Double-click "BaseCamp."  You should get an error message saying you don't have access to it, and if so you'll be invited to click a button to grant yourself full access. 
Once you click that button, you should be able to start BaseCamp again.  It worked for me.