i have a mature bamboo. they do eat the lower leaves (no surprise there) and then they lay in between to stalks and get some shade from above.
Mine is a non invasive type, dont know which one as it was a gift. I think whatever you put in it will get munched to some extent but if you get something big enough at least they will only eat the lower half. They will also dig around the bottom of the plant and probably uproot anything newly planted but you could plant something in a large pot and protect the roots with some large stones
Our coop is on legs so they go under for some shade and our neighbour has a lovely tree that overhangs one corner of the run giving more shade