Fair enough. The only issue I see is the red vest American hunters are required to wear will not go with the full shooting tweed outfit. i suggest you ignore this because you will all stand out enough with the flat cap, plus twos and three piece among a sea of black suits and duck bill shoes.
Maybe you could modify the traditional whistle used to attract ducks to mimic foreign accents? Oh, and you'll need a pointer or labrador to retrieve the prey.
Good hunting, couldn't think of a more worthy cause in one of my favourite cities (even though I am a tourist when I visit, I will dress accordingly so as not to be mistaken for a target).
"Foreign accents" should include those accents from regions of the US due south and west of the Mid-Atlantic States (for myself, I'm just as happy to shoot someone from a red state as from a foreign country).
As for you being mistaken for a "tourist", to the best of my knowledge, you've never presented yourself in oversized, overly long cargo shorts, with a T-shirt and sneakers, and you're certainly not fat enough to be seen as a typical tourist, so I think it will be safe for you to walk the streets of NYC when next you're here - no chance of pot shots being fired at you.