The entire edifice of this "human caused climate change" hypothesis rests on the premise that carbon dioxide is /the/ control knob for this planet's temperature.
There is zero proof that this is true. Zero. So if I were you, I'd be very, very cautious before going ahead with capturing CO2, by any method.
It's way too early for this sort of thing. Vastly more research is required before taking a potentially disastrous decision to try to reduce the amount of CO2 - a gas which Earthly vegetation /needs/ in order to survive!