Well, I'll take a shot at it. Not sure if I can be of much help - but I'll tell you of my experiences. My chickens are in a run, covered with three layers of netting - we just kept adding b/c I said "Wild birds can get in" so I think the last layer is bird netting! Anyway, we built it between a row of existing italian cypress and an orange tree, but no trees in the actual run (about 12' wide). Some of the orange tree branches do overhang the run and there is so much shade in there I am thinking of cutting some back. Plus - it gives wild birds a place to sit in the branches (and poop) and right now I'm wondering how the heck I got a case of canker we're fighting. Lesson learned.
Or, on the other hand, the canker could have come from the place I picked one bird up at, two years ago. They had a tree in the run and had chain link fencing covering it. It was not totally secure to the trunk of the tree, so wild birds could get in and through, and did, sharing their food. So if that is a problem for you, you really need to consider that. You could make the attachment to the limbs flexible by using a little excess barrier (bird netting/chicken wire) that would flex in a wind and maybe secure it with a band or wire..
You have two choices really. One is to build it so it goes over it, like the zoos do for walk in aviaries (which I love the way they do that) - but, seems expensive and an engineering nightmare, or you could attach the wire to the tree with a lower roof. In that case, I'd minimize the # of branches going through the roof to make maintenance easier - just choose two or three. But then, what if the time comes you need to trim the tree? Again, we have had several cypress-type trees overgrow near the run and it is a problem to trim them without damage to the structure.
I don't know what all you are considering, but I think with a tree in the middle you need to look at every possibility - it really complicates things as you probably discovered. I think you are doing a good job of thinking through the problems tho. Good luck - I'd love to see pics of what you finally decide to do.