Have you camped in it in the winter before? There is going to be a lot of condensation in my experience which would make the cardboard a bad idea.
Definitely need insulation under the bed as this is your biggest trap for moisture and the cab over base is not well insulated. When I can afford it I am getting the fruli sleep system. Eliminates moisture under there by allowing air flow. I've heard that reflectix under the mattress works well as does a good closed cel foam. I use my down backpacking mattress in my camper but it gets damp.