[ProctorHillFarm]

My husband runs an excavation and landscaping supply company and probably about two years ago he brought some rubber "pond liner" home- and thats exactly what we did with it! We hung it over the small doors- and even a 10x10 run in, it really makes a huge difference.
We kept ours in solid flaps sort of like a regular dog door- it only takes them about 5 minutes to figure out they can push it open and come and go.

 

[ProctorHillFarm]

we havent had any of ours eat it either- here is a picture of one of our small doors to the barn
its .45 mil thickness- so pretty heavy duty stuff, not sure if it needs to be that heavy though!