Way before my time but what I think the other person confused it with was a train that would leave DC in the evening with sleeper cars. It arrived at NYP in the middle of the night and you were allowed to stay on board until you woke up. This would've been 20+ years ago and not helpful anyways in your situation.
The 3:15am train is a pretty miserable experience all around. In my shoes, I'd stay overnight near JFK or even consider just renting a car one-way if you have the energy. At that hour, you'll probably make better time than the train before you even factor transit to NYP and waiting. The buses do run all night if you go that route.