We got the HTTPS working by using a post-processing rule for WP Engine.
But the biggest problem (taking 2 hours of me chatting with WPE support) was trying to figure out why the thumbnails were being stored in a cache folder. I finally discovered a post about FooGallery stating that this is what you do by default. That was a big mess, maybe you could make it much more obvious that FooGallery is doing this, and how it can be forced to use the actual images instead of cached images.
WP Engine doesn’t by default backup any folders named /cache/ or /caches/ or /cached/, so when I rolled back to an earlier site version all my FooGallery image thumbnails disappeared. It’s simple to tick the checkbox to force it to use the actual image files, but not if it’s not obvious what that’s for.