I like that the Omlet door has a sensor that keeps it from closing on any chicken that's moving through it or poking a head or butt out. It's pricey but worth it.Will they have access to the whole area or just that small run? If they will be in the open area I would string up some hawk string back and forth from the house to the fence.
I recommend the Omelet auto door but it may be too big for your coop as it's a sideways open and not up and down.