Mixed content warnings are built into browsers by design to indicate exactly what they mean. You can turn them off in settings or just click ok, so by throwing the mixed content, you're degrading UI, but not functionality.
A few things come to mind, since the providers can't change their content:
Write a back end scraper for your app that scrapes the web page and servers the content locally over https.
Don't render the content immediately, make the user click on it to open the iframe so that at least your page loads and you can warn the user (optional).
Enhance either solution by checking for https first, a lot of websites have 80 and 443 both open, but as you pointed out, not everybody.
Not too familiar with this one, but you can maybe even have the server instance of internet explorer open the pages and cache them for you simplifying the scrape.
If I was writing this, I would check for https when possible and allow the mixed content warnings as all that's by design.