I use Let's Encrypt as well. I ran into some problems too before I finally figured it out. This fix is for all SSL certs, not just Let's Encrypt.
I did 2 things:
1) add this to your _header file & clear all cache.
<meta http-equiv="Content-Security-Policy" content="upgrade-insecure-requests">
- This will try to load up secure versions of insecure objects on your pages.
2) In your Chrome browser (not sure how to see this same info in Firefox), go to your page, right click your mouse, and click "inspect". This will open up elements coding info. FIX all triangular exclamation point problems (see attached pic). They should relate to what is preventing your page from not displaying in SSL.
Doing both of these allowed Let's Encrypt to work on all my Dolphin sites. It was a pain in the ass for a few of 'em because there were like 7 or 8 things to fix, but it worked.