I had a very similar problem to what you describe, and finally tracked it down today. In my scenario, I wanted to use API Gateway to deliver requests to a private EC2 instance running LAMP stack and WordPress, and needed access to a custom API at https://wordpress.example.com/api/product which is currently only accessible on our private network.

I finally found that Apache HTTPD is redirecting me to a path with a trailing / so it was trying to redirect me to https://wordpress.example.com/api/product/. I changed my Endpoint URL to have the trailing slash, and that resolved the redirecting issue.

Reference: https://stackoverflow.com/a/64533237/2329657