I know that sounds dismissive but I mean it. There's no magic wand.
The proposed carbon storage is permanent in the form of carbonates. This is a variant of the Solvay process used to make washing powder.
Chalk cliffs were once the shells of sea creatures. That's a whole load of sequestered carbon right there.
This kind of battery is useful. All human hearts together produce about the same power in beating as a Falcon 9 in the final stage but you can't use hearts like that.