That would be a dream to make a coop. You don’t have to worry about keeping them warm or giving them a small space their body heat will warm up any more than you have to worry about the wild birds that overwinter needing a cozy home. For many reasons bigger is better.
In Upstate New York with the winters you will have I’d probably section off a corner about 8’ x 8’, somewhere in that region depending on that stud spacing. I can’t tell how tall that barn is but it looks plenty high. Since 2x4’s and other building materials come in 8’ or 4’ dimensions, make it easy and make it 8’ tall also. That way it’s tall enough you can work in there without problems. Since it will be protected by the barn I’d make the top of that 8’ cube wire for great ventilation.
That still leaves you a lot of storage space in that barn. With your winters they will probably spend a lot of time in the coop because of snow. A coop that big gives them plenty of room so they don’t go stir-crazy and start beating each other up.
That just looks absolutely fabulous.