I'm not sure I understand why you want to use API Gateway with FastAPI - in my mind there is a lot of redundant functionality there that means you're building/paying/using two things to do the job of one. What benefits does using FastAPI give you that you don't get using API Gateway?
If you're looking for a lightweight way to host containers then I suggest that App Runner might be a better solution here.
How to add python libraries to lambda using the CDK?Accepted Answerasked 2 years ago
Python Fast Api deploy in AWS lambdaAccepted Answerasked a month ago
Lambda response compression to API Gateway and clientasked 5 months ago
ASP.NET Web API on AWS Lambda CORSasked 4 months ago
Authorization Lambda caching based on api-key and pathasked 5 months ago
Deploy Lambda Function and API Gateway RESTasked 5 months ago
AWS Lambda Layers with Python Packageasked 6 months ago
Binary uploads to API Gateway Proxy with Lambda IntegrationAccepted Answerasked 5 years ago
How to use AWS CDK to compile and deploy a typescript api with dependencies to lambda?asked 6 months ago
AWS API Gatway to upload File To S3 without AWS LambdaAccepted Answerasked a month ago