Your cooler will gain heat from:
* hot air around it, conducting
* hot ground it's sitting on, conducting
* hot air/objects, radiating heat to the cooler
* sunlight, converting to heat on the cooler (why white works better)
If the cooler is inside a hexayurt: no sun, limited not so hot objects, air is cooler than outside the hexayurt and the ground is much cooler than the ground outside (less so right beside a sun-ward wall).
In fact, the ground should be cooler than the air inside the Hexayurt (if not beside a sunlight wall), such that the cooler sitting on the ground should absorb less heat as any heat contacting the outside of the cooler has a chance to conduct away into the cool ground (60F?). By all means use insulation on the sides and tops of the cooler. The more the better, but you're already in the best possible situation for the cooler. If your floorplan allows for the cooler to be on the North side of the hexayurt, I'd considering not having insulation between the cooler and the ground.
I often wondered if a small box, not tight fitting, of the silver foam core board made to surround a cooler would improve cooler performance meaningfully. If in a tent it certainly would; if already inside a hexayurt? And without a bottom if inside a hexayurt?