You really need to keep the temps inside the grill below 100, so when you're starting at 96 it's probably close to impossible. The harder the cheese the higher the melting point, but that's still going to be tough.
Before I had firebricks to isolate the hot coals and wood, I used to smoke cheese on a cool winter night and when the cheese started to get too wet and shiny (starting to melt) I would pull it off the grill and sit it out in the cold too cool off. You could try moving them to a cooler or a freezer to harden and smoke in short intervals, but again that's going to be tough without melting it. At least in Houston in August....