But I really want to do a synthetic check whether the proxy is buffering (like promised) the output of the application server (which resides on a different machine).
Maybe I'm missing something, but can you simply check the web server logs to confirm the request came from the proxy instead of the endpoint? And/or check the proxy logs (perhaps increasing the log level for server connection/caching details)? That should at least answer that question. Most proxies also allow you to customize HTTP headers to some extent, so you can even stick some metadata in there to help get more objective results in your testing.
For a very simple "slow client", the LWP::UserAgent callback suggestion is reasonable for a starting point.