maybe you have some kind of natural evaporative heatsink nearby (trees) or perhaps inside

Is it the North side? Maybe painted a different color?

That's your lower bound, but you need to have fans everywhere to replace all air and cool down your walls to the air temperature.

True, but we're at the time of year when there's way more daylight than night, so it's a war of attrition with the hot day winning no matter what you do.