That is fundamentally what a calcium reactor does. However, the calcium doesn't disolve in normal seawater, it needs a lower PH. This is why CO2 is pumped into the reactor. It dissolves into the water, lowers the PH, and the calcium dissolves into solution as Ca (calcium) and CO3 (carbonate) ions.
So, really, you need to modify your reactor to accept a way to dissolve the CO2 from a CO2 tank into solution, and mix that through your calcium.
If you look at my reactor (which I can't even remember the name of), it takes the CO2, pumps it through a pinwheel impeller or a recirculating pump, mixing it into the water. So, the water flows around the chamber, distributing the disolved CO2. Then, a small pump pushes water at 5g/hour through the chamber, back into the tank.