This is what I remember from working for a start up that was helping make low carbon cement, the numbers could be wrong, but it's more complicated than just the energy to reach the high temps.
Then after grind them up as mulch abd fertilizer.
We can outfit the bacteria with a natural inhibitor. Maybe a lack of lysine.
Yeah. That'll work.
The obvious economic quirk about capturing carbon in concrete is that given time and pressure you can convince carbon to harden. Concrete that continues to pull in carbon in its process in theory has good chances to be stronger and hardier. If it continues past the initial set, you get "self-healing" concrete.
You joke about plants, but yes plants have been doing variations on that for epochs now, there is probably some more lessons to apply from them if we want to build a "diamond age" with all the excess carbon we've spewed as a culture.
The reason I diverged from vines, is because multi-cellular is too slow. I think what we'd need to do, is create some sort of thin mat you lay over the concrete, just unroll it, and it has dormant bacteria along with whatever input(food) it requires to consume CO2.
So it then, much like yeast in bread, multiplies quickly, and you're left with a thicker mat to roll up and dispose of, grind up, say for fertilizer.
I envision the mat to be thin, akin to plastic rolls you get to keep insulation in place. They could be comprised of entirely organic materials, food for the bacteria. You know, such a thing should be possible, but would the bacteria multiple quickly enough? And likely water would be required as an additional.