I did something similar with https://tinysite.adamdrake.com where I have a website that fits into a single ipv4 datagram (at least it did when I was using my own HTTP server).
When I moved it to AWS Lambda there was some size introduced from TLS and also from some headers that AWS API Gateway adds which seem, unfortunately, impossible to easily drop (e.g., if you make the headers empty in the response, APIGW will copy them into another header and still add its own).
I might pull that service off of AWS APIGW/Lambda, in which case the size as an artifact of AGIGW would disappear again.