Very nice!
You could also do this simply by running on Azure Functions (free consumption plan). All you need to deploy is a simple proxies.json, see [1].
On the AWS side it appears you can entirely use API Gateway. Here [2] is a Swagger 2.0 definition file I just wrote which upon import creates an API endpoint that should proxy requests to an external API while adding your secret credentials.
[1]: https://aka.ms/apiproxy
[2]: https://gist.github.com/berndverst/b59fa0b38544f13c05ae0b77f...