Chris, you were close... the dialogue box that it's trying to display is the box that asks if I would like to "help improve RedGate Software".
I did check the registry, and add permissions as you suggested. I still encountered the same problem.
In order to get this error to disappear, I had to log into the server as the service account that ends up executing the stored proc, and manually run SQL Data Compare from the command line, and click through the "help improve RedGate" pop-up. Now it will work when executed through the stored proc.
Although I did click "No" on the "help improve RedGate" prompt, I do have a suggestion...