Brenden, do you mean to use Functiona Caller and write a custom client code for the REST service, and then wrap an API layer around it using the Webpart to API piece?
Just to clarify, we're only looking to inject some data from a REST service into a page before being sent to the client. It would be a great if some existing web part can be configured with the REST endpoint and then render the response of the REST service into the page via transform.