The Lambda Handler to proxy any other API Gateway/HTTP endpoints. This accepts only Authorization header as the request and adds CORS headers to the response. No dependencies.

lambda-cors-proxy.js

const https = require('https');

exports.handler = async (event) => {
    console.log(event);
    const [res, body] = await new Promise((resolve, reject) => {
        const req = https.request(`https://example.com/${event.pathParameters.proxy}`, {
            method: event.httpMethod,
            headers: { 'authorization': event.headers.Authorization },
        }, (res) => {
            let body = '';
            res.on('data', dat => body += dat);
            res.on('end', () => resolve([res, body]));
        });
        req.end(event.body);
    });
    return {
        statusCode: res.statusCode,
        body,
        headers: {
            ...res.headers,
            'access-control-allow-origin': '*',
        },
    };
};

Setup Lambda Proxy Integration as ANY /{proxy+} with this Lambda Function.

https.request(..., {
    headers: event.headers,
    ...
});

didn't work. I guess because of API Gateway special headers like apigw-requestid

Note that some AWS internal endpoints don't accepts Transfer-Encoding: chunked body.
Below sends chunked body:

req.send(body);
req.end();

Below does non-chunked body:

req.end(body);