To deploy, create a Lambda function and enable function URL (no auth - yolo), then use the handler above in your function. That same implementation will also work with API Gateway HTTP (aka v2), if you want to use ALB or API Gateway REST (aka v1) you should swap the schema used for parsing.
Then you can test using a POST request with this body:
{
"jsonrpc": "2.0",
"method": "tools/list",
"id": 2
}