As far as evening out the temperature, your mash will equilibrate temperature wise without stirring. Stirring will speed up the process at first but once you grain is saturated with water, the temperature will be pretty uniform throughout. I would agree with your thinking of temperature loss when opening the cooler as a much more crucial factor. Stirring might help the starch conversion speed but probably not a substantial amount. So I would say don't stir once your grain is saturated with water and your target temperature is reached.