Live Preview and CDN caching
If it were me, I wouldn't use Cloudfront to offload the entire site. You can certainly use it for static assets but cache invalidation is really expensive. And as you've found it, troublesome. By offloading the static assets, you can keep the web server focused on just serving page requests.

From your previous question, it sounds you get a fair amount of traffic. 

Cache invalidation is a tricky subject. I don't know how often you update but a middle-of-the-road approach is using nginx or varnish.

I like nginx myself because of its flexible fastcgi_cache. Even if you set the cache low, like 1 minute, you've just increased your capacity by an order of magnitude without much additional work on your part.

nginx also allows you to cache by location so you could exclude caching any page with a form. POST requests by default bypass the cache.