Cat will make the decision on occupancy of the cooler/shelter.
I have visions of Snoopy sleeping on top of the dog house.

Suggest that you NOT use straw. Hay is better choice.
Remember reading that dogs could chew straw and it could perforate intestines.

Intelligent Donkey comment - (using synonyms to avoid infraction) :
One large round hay bale - reach in and pull out some of the center - leave a perch out front about half way up for cat to use for access. (five gallon bucket with lid comes to mind) well sheltered and well insulated.

Bekeart