this is a very detailed and thoughtful question!
We have 2 or more chambers, we are balancing the number of chambers so that a chamber is always adsorbing while the other one (or 2) is transitioning temperature or holding a hot cycle to regenerate.
We adsorb at ambient temperature so cooling the previously heated device is not too much energy, mostly just a fan, but there is some available cool refrigerant from our dehumidifier/heat pump cycle that removes moisture from the exhaust to help quickly drop the chamber temperature.
We can use the exhaust waste heat to start heating the bed that is going into the hot regen cycle, then we use the waste heat from our exhaust dehumidifier to get the bed the rest of the way to the target hot cycle temperature.
Its hard to explain in text, but you can see that we get to use the hot and cold cycles from the refrigerant cycle to do work for us simultaneously and take advantage of the heat in the exhaust itself!