AWS SAM Templates - API Gateway with multiple stages and proxy to URI (NOT LAMBDA)

I find different pieces of the puzzle I need to solve, but none of the examples I've found online show how to do multiple stages in a sam template (ie: Deploy dev, prod, and test) at the same time. And none show how to proxy to a URI that I've found. All of them show the gateway going to lambda as a proxy. But how do you do a standard http_proxy in the template?

Sorry, new to SAM so trying to figure things out. I need 1 API gateway with multiple stages that http_proxies to a URI. Each stage goes to the same place, but different API keys used for each one as a stage variable. And I need it to be private with two VPCE's so I have to attach a resource policy to it as well. All in SAM :)