While I cannot profess any knowledge of the current NYC laws (sorry), I cannot see a barrel alone being a problem, as it is not a firearm.
However, NYC has a reputation for not being "reasonable" about their firearms laws, even to the point of not accepting permits issued in NY, outside of NYC. (lived in NY state for 18 years, and this was the case then).
If you are going to store the gun, seems foolish to keep the barrel with you, to me, anyway. The odds of the barrel being lost, or damaged during moving (or after), or even stolen while you are living in NYC are irrelevant, IF you don't take the barrel with you. Also, if you store it outside NYC, their laws don't apply.
My advice is store the gun, and all barrels & accessories, in a secure location. Someplace covered by insurance, so that if the worst happens, you can at least get something back.
All else being equal (and it almost never is) bigger bullets tend to work better.