You can absolutely pass through the state under federal law. There have already been some cases dismissed and a precident has been established but that doesn't mean a LEO won't initially arrest you. But the same holds true for most states, not just NY. Once the facts are established that you are passing through the charges are generally dropped. But!!!!! You now have an arrest record... not good.
NYS PL S265.20(13-a): Exemptions - allows for a non-resident to enter NYS with a handgun under certain circumstances. The NRA link fails to mention one of those circumstances. you can enter NYS for purposes of training provided you are taking an NRA certified class, have proof of registration, have your home state FOID or CCW license ane have locked your unlloaded gun in the trunk. You may enter 48 hours before the class and must be gone within 48 hours of class completion.
The actual wording is:
Section 265.20 Exemptions
a. Sections 265.01, 265.02, 265.03, 265.04, 265.05, 265.10, 265.11, 265.12, 265.13, 265.15and 270.05 shall not apply to:
13-a. Except in cities not wholly contained within a single county of the state, possession of pistols and revolvers by a person who is a nonresident of this state while attending or traveling to or from, an organized convention or exhibition for the display of
or education about firearms, which is conducted under auspices of, or approved by, the National Rifle Association.
I go to NJ for training all the time. It's like a deoderant commercial... see how much I sweat.