If your ambient room temperature is much over 38C then a cooler would make sense, if it is able to keep the temperature consistent. Bear in mind that the first developer is in the tank for only a couple of minutes, so an alternative might be an insulated drinks-box containing water bath, chemical-bottles and dev-tank, all at the appropriate temperature before starting processing. The temperature will not change during the critical fist developer, thereafter change will be minimal and could be controlled by a putting bottle of frozen water in and out of the insulated box. Using a similar method for warming the chemicals (rather than cooling them) is fairly trivial and works perfectly adequately here, so it should be fine the other way round too.
Edit: I just re-read your post about moving the containers from cool to warm and back again. Why are you doing that? Just set up an insulated bath at the right temperature, place the containers in it and allow the whole thing to get to the same temperature (with adjustment to the complete environment, not individual containers).