The is nothing in Servant to do this out of the box, however all the needed parts are available.
Before we begin, I guess that if you can stream to a Sink, that means you have a source (the gorsBody of GetObjectResponse is a RsBody, which is a Source)
First of all, Servant provides us with the possibility to add support for new return types, by creating a new instance of HasServer, so we could serve a EitherT ServantErr IO (Source ...) and have it stream.
To create that instance, we must implement route :: Proxy layout -> Server layout -> RoutingApplication. Server layout, in this case, just means EitherT ServantErr IO layout, layout being the source we want to server, so it's the function that returns a source (and may fail with an HTTP error).
We have to return a RoutingApplication, which is (in continuation style) a function that takes a Request and returns a RouteResult Response, which means either an unmatched route error, or a response. Both Request and Response are standard wai, not Servant, so we can now look at the rest of the ecosystem to find how to implement it.
Fortunately, we don't have to go far: Network.Wai.Conduit contains just what we need to implement the route function: responseSource takes a status value, some response headers and your source and gives you a Response.
So, that is quite a lot of work to do, but everything we need is there. Looking a the source of the instance HasServer * (Get ...) might help.