Go buy a $15 bathroom ceiling fan at Home Depot, install it at the top of the stud bay in the heated room, and place a vent in the wall at the bottom of the bay in the cold room.
Better yet, instead of the $15 fan, get one that has a very low noise rating (a quiet one) and make sure it moves A LOT of air. Still should be around or less than $100. You could even put it on a thermostat or a timer, or a plain old wall switch.
Dunno, seems like it would work fine to me... Make sure you don't have any fireblocks in the bay or this would do no good.