Back to the orginal question. Storing heat will be a bit of a problem. (transporting such a mass may be a big problem). There are things like this:
https://www.amazon.com/HotSnapZ-Reus...8243843&sr=8-8
they do a phase change. I am not sure of the thermal capacity of these at the moment.
Another aspect of climate control is humidity. Humidity has a strong effect on how you
perceive temperature. A dehumidifier can be recharged using excess exhaust heat. That would be a worthy use of exhaust heat...
I personally was thinking of using the exhaust to distill water.