Your presumption is correct. Use a different bowl for the sponge.
Base the coverage of the top screen on your temp and humidity reading. Start with it covered and move from there. If the humidity reads correctly but the temp gets too high, use a lower wattage bulb. You want to use the least amount of cardboard you can to achieve the desired results. You want to get fresh air in the tank too.
Not sure where you are but this is something I did in the winter and during sheds. The humidity was ok in the summer.