From what I know from my own chickens, they would likely be way too chicken to shove through a doggie door. BUT what you could do is to lock them in at night (Which is best since predators are much more likely to use that door than the chickens anyway) and simply prop the door open for them in the morning. If the door didn't come with the locking panel you should be able to make one out of sheet metal. It should have a second door that drops into place that would keep the pet locked in or out.
I lock my chickens in every night and let them out each morning for safety from the local Possums and raccoons.
I found that when I lined the inside of the barn door with a layer of plastic that you use to protect furniture from paint splatter that my chickens are scared to death of that stuff and it is thin as paper! Every day they wait behind the liner while I roll it up and clip it up out of their way, then they proceed outside to play. It's way thinner than doggie door plastic!