I don't know about differing insulating qualities. I used straw because it was much cheaper than hay.
I had a cat-house outside insulated.....made it out of a plastic tub you store things in so it was wind proof as well...plus you could just take off the lid to change the straw. Like yours it really stayed warm. But I do understand straw is better at insulation...don’t know why though.
I made it because sometimes I couldn’t always get to the door right away to let him in...in fact sometimes he wanted to stay there! LOL!
Had to put him down couple months ago or so....he had a good life here after I had rescued him.