If you'll recall from my post Abstracting WCF Service Calls in Silverlight, it is relatively straightforward to build a service and reuse it in a way that you can install the application anywhere and have it function correctly.
I failed to include some extra steps that are required when the service is secured via SSL. I knew that the bindings would require extra information in order to facilitate this, but I wanted to see if there was a simple, standard solution for doing it.
Glad I was able to find the solution here, which makes perfect sense: enable HTTPS as a transport option, then set it on the Silverlight side where applicable.