Running a Fargate service behind an API Gateway. You get all the advantages of API Gateway (e.g. Throttling) and it might be cheaper as an ALB (up until ~16 million requests per month)
The picture shows more than what I have in mind for the pattern. I'd focus on API Gateway / CloudMap / Fargate.

source: https://aws.amazon.com/blogs/architecture/field-notes-serverless-container-based-apis-with-amazon-ecs-and-amazon-api-gateway/