[Wharf Rat]

First, I agree with Bobarian.

If you insist on trying this, consider: Let's say you start with air in the cooler at your room's humidity. As you cool it, the relative humidity will increase, so the humidification control in your humidor will be challenged to keep up. So, you may be controlling the humidor's temperature at the expense of the humidity.

I think it would be better to just put the humidor in the cooler without the ice. This will slow down how quickly the temperature and humidity change.