internal door question

In my kitchen hallway is a wooden door to my basement and then 4/5' from that door is another door to the outside of the house. My basement is very cold and drafty, and I would like to find a door in the style of the house(5-panel horizontal, craftsman style) that is insulated. I would just get a fiberglass external door and put it there however my wife doesn't want to have a plastic looking door inside the house.
I heard that I might be able to get an mdf door that was also better insulated than just a wood panel door but have been unable to find something.

To make matters more complicated I need to put a cat door in this door as our cat's litter box is in the basement. Currently in the wooden door that came with the house(prolly 1920's) I cut a hole and put in a cat door.

I'm open to thoughts on how to make this area less drafty. Granted replacing the exterior door and storm door that is near the basement door will help a bunch, but I still feel I should replace the interior basement door with something in the house style that is insulated.

Ahh the wonders of a 1920s house in new england!

Thanks!

maybe fix rigid insulation to the back of it? then cover with hardboard, mdf etc;

I think I would have better luck convincing the wife to let me use a fiberglass door there over having a wood door with foamboard affixed to the back of it.