The API responses are returned in gzipped format. But by using Content-Encoding gzip, the burden of unzipping is on the user of the API. On the other hand if the server uses Transfer-Encoding gzip, many http client libraries (for eg. Python's urllib) will do the unzipping automatically. The server gets the same bandwidth savings.
It's the same as much of the web.
As a rule, you don't want
What you want is the actual content, exactly as it would be without the content-encoding, but to get it faster. That is to say, you want
So all in all, Transfer-Encoding is the blatantly obvious choice both here and in just about every other case where Content-Encoding is used.
However, the support for Content-Encoding is vastly greater than for Transfer-Encoding, with tool-kits, apis and libraries for both client and server giving support for the former vastly outnumbering those that do for the latter, which results in a spiral - it's not worth servers supporting it when clients don't use it and you can kludge it with Content-Encoding, and it's not worth clients supporting it when servers don't use it and you can kludge it with Content-Encoding.
Which makes it a case of "if all the webservers, browsers and web-client toolkits jumped off a bridge, would you do it too?" To which the answer is "since we have to interoperate with those toolkits, yes".
Transfer-Encoding is hop-by-hop, while Content-Encoding is end-to-end.
This means that if there is a proxy involved, anywhere, the proxy will see the TE gzip, unzip it, and not necessarily forward the request as TE gzip.
So, the choices are
The logical choice is CE gzip.