I was able to identify the problem by cutting and pasting the raw HTML source of the page into a new page (copying chunks of HTML code one by one) and testing for when the SSL-certificate-validation broke.
It turns out that there was a broken image path for a 1x1-pixel shim gif. Once I corrected the image path, the page displayed correctly (no "insecure content" error).
It's curious that a missing image would cause the browser to display the "page contains a mixture of secure and insecure content" error, but hopefully this will help someone else in the future.