Sounds like a couple issues in the mix: 1) interstate transport under federal law, and 2) NY regulation of possession/storage of long arms.
1) FOPA's interstate transport provision may cover you for a "just passing through" stay in the course of interstate travel
if you are legally allowed to possess the firearm at
both your
destination and your
starting state. But you're STAYING in NY on vaca - so FOPA really applies to the states you travel through on your way there. So in short, NY law has to be looked to since it's your destination. If you can legally possess the shotgun in NY under
NY state law (and where you are starting from), then under FOPA, you can transport it locked in the compartment as you say.
2) I don't think NY State requires a permit for possession of long guns, and a pump shotgun transported locked (and unloaded) in a separate compartment works for most states' transportation and shouldn't fall under NY's AWB-type provisions,
but I am not a NY lawyer, so I'll leave that question to others expert in NY law. http://www.nysrpa.org/nygunlaws.htm is one place to start. You WILL need to comply with all NY State laws regarding carrying, possession, etc. with regard to the firearm while you are in NY State.
In short,
if you can possess it legally in NY state (and your starting state), you can travel with it there in the way you say.