Unpopular opinion: AWS lambda is basically equivalent to CGI. The web has come full circle.
That said, building a FCGI "bridge" isn't hard, it's just that nobody cared enough to do it.
Basically: your instance POSTS to an endpoint to get/reply to requests, one at a time.
I can't wait until you can specify that a lambda instance can handle a certain # of requests in parallel. Ex: things just blocked on IO/DB. That would make better use of your RAM.