SSL Issues with Mixed Content and Firefox
When you set $base_url, Drupal always uses it in email absolute links and whenever it has to build an absolute URL. 

I have many sites running full-HTTPS, with only the $base_url set to the https URL. 

We can't debug your site for you, but check the page source to see if any Javascript files are being served via HTTP. 
Google Analytics, Google fonts, Google translate widgets, banners, ad serving scripts, CDN-served jQuery, etc. 

Header tracking and other debug features won't probably work in your browser because when the page URL is in HTTPS, browsers do not load HTTP assets. You can still check the page source to see the HTTP references.