As a former Manhattanite myself, I would vote for using one of those grocery/laundry carts, the bigger size. Stick your BOB or other supplies in a laundry bag and no one will be the wiser.

Perhaps looking homeless will be the most "invisible" but as long as you don't look like the investment banker or Madison Ave hipster, you shouldn't draw much attention pushing one of those carts down the sidewalk. Probably not common in many cities, but you see so many of those carts in NY, no one will think it looks out of place.

Plus, you can certainly carry more weight for a longer distance in a cart, which will be important if you have to hoof it somewhere, like out of Manhattan across one of the bridges. If you really wanted to take the pistol-grip-only or folding stock shotgun with you, the cart will also allow you to hide it more easily/conveniently than on your person.