The Blue Planet System directly converts CO2 to carbonate for mineralization as a synthetic limestone. In other words, Blue Planet’s technology uses CO2 as a raw material for making carbonate rocks. As a result, the carbonate rocks produced are used in place of natural limestone rock mined from quarries, which is the principal component of cement.
So, what you are looking for is the cost of cement. As long as I remember, the cost of carbon-capturing of cement is around 60-120 $ per tonne.
Carbonation makes concrete more hard but brittle. Basically , carbonation process is slow . For carbonation activity, moisture is essential. If specimen is kept in contact with higher concentration of CO2 with moist environment, the rate of carbonation can be accelerated. However, achieving the carbonation at large depth may be challenging. Effect of temperature is need to be studied.
Please also remember that carbonation reduces the pH of concrete, and so any form of steel reinforcement will corrode, even in an interior environment.