We have no crawl space in our living room in sw fl. No place for insulation. Ceiling beams are attached to plywood attached to shingles. Ac runs constantly but doesnt cool the room. The rest of the house stays cool. Two ac units. Will an attic fan work to pull the hot air out of throom, very high ceilings(20 ft approx), without pulling the cool air out.
Consider installing a drop ceiling and adding insulation on top of it.
NOTE: Cold air is heavier than warm air and should stay at lower level of room, however an exhaust fan could drag it out. If you were going to try it.. In this case ...SIZE definitely matters! However, I will defer to the pros out there. especially those with Southern experience.