The routing codes for this would be "LAX JFK X?" (Meaning, connect in LAX, then JFK with no further connections) but submitting that shows only DL and AS options in Matrix because there is no non-stop AA flight from JFK-SYR. You could use codes: AA+ LAX JFK DCA and AA+ DCA JFK LAX for the return but those really, really terrible routes and prices out at 2786.00.
if you really want a bed for the transcon you could do DL and its shorter but still a 20+ hour journey and its super expensive:
So tl;dr: take the recliner.