Hi Dave,
Sorry for the late reply. My subscription to this thread does not appear to be working for me. I just realized that I was talking about product ESRI SilverLight, not the Sharepoint webpart. I really apologize for the confusion.
But, my point is probably still valid. To troubleshoot the problem I would suggest using a tool like Fiddler or Google Chrome developer tools. These tools will allow you to see all the network activity for every post. In the logs you will probably see SharePoint trying to GET an image from an HTTP url, instead of HTTPS. If you can fix that url so that it is HTTPS, I believe your problem will be fixed.
-_Rick