How about some Republic Airlines service? AA offers it on JFK-DCA-JFK and it is lovely. It's a nice compromise between mainline and Eagle. No need for valet checks even!

Ironically, AA also offers 1x daily mainline service on JFK-DCA-JFK (mid-afternoon outbound, late-afternoon return), and it is a shorter route than JFK-RDU-JFK! Based on the timing of the segments, I think it might be to allow the Washington bigwigs to fly mainline when connecting through JFK on their way to/from Europe...