I've used thin foam boards in many applications. Attached to the inside of your cabinet door and trimmed out with foam backer rod ( comes flat and round) might work. Finding an adhesive that holds the backer to the foam board is the trick. This can be close to airtight but not 100%. Sometimes the backer can be attached to the inside frame for a good seal. Usually the door's hinge side is a problem but the hinges can be shimmed out. I'm assuming you want to keep using the cabinet door.
