AA operates the Shuttle DCA-LGA. It's far from the pre-pandemic hourly service, but still multiple frequencies. An easy Uber over to JFK. Presuming that you are originating in DC and not out by IAD, the combination of the ease of getting to DCA and the simplicity of the airport may make up for the hassle of the Uber. You can also fly DCA-JFK on DL for $149. No chance of a reroute, but that price might make the hassle reduction worth it.