Turning a dowel or tube to wrap a cord around over many full turns of the motor vs turning an arm on a frame through an arc with insufficient leverage for the power available and frame blocking the door issues... and which is easiest? Does that include to live with?
If you own the door, you can mod the door like people already modify chicken coop doors. Some are real good, counterbalanced so open/close don't take much power. One has a circle with a hole to match the wall hole, he turns the circle to open/close the hole. All he needs is a board wide enough to cover the hole and twice long enough to cover it from a rotation point above the hole so the top and bottom balance each other the motor doesn't have to lift, just turn. move the board away from the hole, pet goes through. If the bottom weighs just a bit more than the top you can get return to closed w/o a spring, open by pulling the top to one side (wind cord onto spindle) and close by unwinding. If someone opens the door, they open the whole door.
IMO it needs sensing to make extra sure nothing is in the hole when the cover swings shut and some kind of collar key to identify pets from "animals".
I have some dead/useless printers to salvage. They have some hefty looking steppers, a lot bigger than pulls from 5 1/4" FD's.
But for what you're doing, wouldn't it be easier and cheaper to drive a DC motor especially as power needs increase?