Very nice article, I went for a little bit of different solution when I had to solve this exact problem. I put together a post of it (link below).

I put the DNS-update logic inside the container so that it self-registers on Route53 with no need of external logic in Lambda etc. https://itnext.io/getting-a-persistant-address-to-a-ecs-fargate-container-3df5689f6e56