How to get cold air from one room to another
I have used temperature controlled fans in the past with a covers to transfer heat from the lower floor where there was a high efficiency wood stove to the upper floor. You could do the same thing in the wall using return air grates to cover the fan, the most expensive part of the project was the fancy window fan I used that both low and high temp cutouts and could vary the flow by using 1 or both fans. I don't. Remember the brand but it was expensive ~$125 for a fan but it has been in place for 14 or 15 years. I did put an outlet in the cavity so it looked like a return air vent in the downstairs and fancy brass heat registers upstairs I think it looks great and it moves the air quite Well, I think this would work in your case also.