Where on the shirt do you want to attach the pocket? That can be critical if, as you say, you plan to carry 1/4 pound in it. You alluded to a kangaroo pocket - which would not interfere with the shoulder straps and will swing less. It should be located fairly high on the shirt, much like a regular breast pocket on a man's shirt.
The silk is likely to sag out of shape with 4 ounces on it. Your idea of supporting the pocket sounds good. You might consider running a couple of light ribbon supports to the shoulder seams adjacent to the neck opening. These could be on the inside.
I think bug net would be strong enough and as you say, it would breath, although the items in the pocket, such as a map, may not. No-seeum netting weighs about 0.9 oz. Thruhiker has some that is even lighter. One consideration with net is that it may not stretch as much as the knitted silk. I would stitch it on with a zig-zag.
Thread? Regular polyester. Silk thread is not very strong and can be hard to handle on some machines.