They are planted inside the run. Yes, they do weigh a bit. A couple years ago I weighed the winter trimmings from all three vines (minus the leaves) and the total was just under 50 lbs. I'm not familiar with the material on top part of your fence to know how it would hold up.
I have experience with some of these plants. I'm not sure about the potato vine. I know it's not the same, but my chickens have recently grown to love sweet potato vines. After years of ignoring them they suddenly decided they are tasty. I don't have any personal experience with the honey...