Dans, I would use straw in the tunnel as you can then compost it and dig it in. Have you thought of taking up the weed suppressing fabric while the geese are in there, so their droppings can fertilise the ground. They will eat any weeds that come through.
Mab I agree about not trying to feed hay to hens. They have no way to break up long strands. Mine are now about 2/3rds of the way through a giant cabbage, although they won't touch the sprouts and kale. Soon chickweed will start to grow and they love that.