This is a little different from what I'd been thinking before your message, which would be more of a fluidized bed reactor -- like quicklime swirling around in air in a garbage can. I think that's closer to how some powerplants treat exhaust.
I've read that there are microscopic changes to the quicklime particles after a couple absorbtion/heating cycles (their pores fill up or something), after which you need to do something to further refresh it, maybe by dissolving it and precipitating it back out of solution.
This led me to considering whether aqueous chemistry might be better.
The simplest would be to react air with calcium hydroxide (would you use a bubbler? Or a packed counterflow tower? Or a spray tower?) and collect the calcium carbonate that precipitates out.
But, based on my shaky chemistry, I understand solubility of calcium hydroxide is low, so you do better with a two-stage process like the Kraft Process:
1. Absorb CO2 in contactor:
2NaOH + CO2 -> Na2CO3 + H2O
2. Regenerate solution in causticizer:
- CaO + H2O -> Ca(OH)2
- Na2CO3 + Ca(OH)2 -> 2 NaOH + CaCO3
3. Decompose calcium carbonate:
CaCO3 -> CaO + CO2
The concentrated NaOH solution would be very caustic, so you'd have to be careful with that, but apart from that none of this seems too terribly scary?
Some parts of the above are endothermic, others exothermic, so maybe this could even let you shift energy from the summer (solar furnace?) to the winter when you need it, as part of the deal.
Or maybe this is all too complicated and you should just use a canister of soda lime granules, like an anesthesia machine? ( https://en.m.wikipedia.org/wiki/Soda_lime )
I'd be interested to hear more from people whose chemistry is better than mine.