Foil up is OK. If the foil was down, it would probably be the same because the interference would go through the cardboard.
You must understand that your situation is unique, and you should be prepared to do some experimentation to see what works.
If I were there with you, I would want to try as many ideas as I can think of with cardboard and a roll of heavy duty 18" aluminum foil.
If you want to get fancy, you can look for aluminum foil radiant barrier sheathing or AtticFoil, which is probably available in your area.
https://www.google.com/search?q=alum...iw=911&bih=345
AtticFoil will even send you a free sample:
http://www.atticfoil.com/