Just a note on adding insulation. I feel the highest priority for insulation is on the bottom of the box. Heat goes to cold. In your house the biggest difference between the inside of the house and the outside is at the top of your ceilings since the heat in the room is rising to meet the cold air in the attic. So insulation in your attic is going to pay bigger dividends that say your walls since that is where the biggest temperature differential is.
With the box it is the opposite. The cold air in the box is dropping to the bottom so the biggest temperature difference is going to be there with the warm air under the box.
With that in mind I'd work as hard as possible on more insulation at the bottom of the box either inside or out or both and of course if you can add to the walls do it,
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