I seen a sleeping box design they use for sled dogs in alaska etc
The way they design them is have 2 doors the normal 1 but then have a partion right at the main door bu the 2nd door is tight to the front ( right beside main door) rather than the back where the most obvious place to put it would be.
Idea being any draft wont bounce of the back wall into the sleeping box.
Could u try a divider inside the sleeping box like that or probably easier a solid wall a few feet in front of the door so dog has to zig zag or enter daigonianlly but should stop most draughts.
I tiled my sleeping boxes, althou easy for cleaning bloody hard and cold even thou well insulated below.
I used to have rubber matts down with vet bed on top, but the rubber doesnae let the water away.
Now went to old astro turf and really like it, its the proper 3/4G stuff of football pitches so the 'grass' is a few inches long, lets any moisture away from the dog and nice and soft with vet bed on top.
Can buy it cheap as chips in big rolls, quita few dairy farmers use it for tracks in there fields for cows to walk along, think about 1T or so as a 3t digger really struggles to lift it, most of the weight is in the rubber stuff they must spread on top of it, takes a bit of getting it all out