Two ideas, and I'd do both.
1) Have HC up the chicken side of the fence high enough to protect the plants.
2) Have a second chicken wire fence on the outside of the fence for the plants to climb. Make it a "lean to" fence that starts a few (4-5?) inches away from the main fence so that the plants can get a good start climbing before the get close enough to be pecked at.
Chickens ripping off a wandering leaf that ends up in their run shouldn't be enough to damage the plant once it gets growing. This could be a nice bit of shade, and a source of food.