In order to check in for the Delta flight as a non-revenue passenger you will have to clear immigration and customs and then proceed to the Delta check-in counter. You will also have to pay the departure tax. Then you only go to the gate after you get the boarding pass at the check-in counter. Not the answer you were hoping for but the way it works there. Note that sometimes even with seats available non-revs do not make it because of weight restrictions due to heavy cargo loads and high temps. The good news is that after the JFK flight you get another shot on the Atlanta flight which leaves later. Good luck.