For something to be useful as an energy source, there must be a reaction using readily available reagents that provides more energy out than it takes in.
Unfortunately, CO2 is essentially "ash," as in the waste product of an energy source.
Although plants do take in CO2, they use it as a raw material to create an energy carrier, which is different from an energy source. In the case of plants, their energy source is the Sun, which they use to split water and attach the resulting hydrogen to the CO2 to create carbohydrates (sugars). These sugars serve as the "battery" which partially stores the energy they've received from the Sun. Later, the plants will take in oxygen to burn these sugars, and release CO2 themselves in the process. However in the case of plants they are generally able to take in more CO2 than they produce, which is why CO2 is such a good fertilizer for plants, and growing plants is a good means of carbon sequestration.
Animals come along and eat the plants to steal their "batteries," (sugars) and thus they can get more energy out of the plant to power themselves than it takes them to eat the plant, and give off CO2 as a waste product (which the plants then pull in again to make more sugars). Some of those plants also die and get buried, where their sugars undergo reactions that eventually turn them into other energy carrying molecules that apes will fight over to maintain the lifeblood of their comfortable lifestyles.
CO2 is an extremely stable molecule due to its pair of double bonds, and it doesn't really like to react with a lot of things under normal conditions. So, unless your world has something that can allow you to react the CO2 and get more energy than it took to create, you won't be able to use CO2 as an energy source. Remember that ultimately, even though we use fossil fuels, our civilization is powered by the Sun. It's just that that energy has already been captured for us; we're stealing batteries that were already charged millions of years ago, and the result of releasing that energy is a waste product that is already in its lowest energy state. Above all things, the laws of thermodynamics cannot be violated. It will take more energy in to do something with the CO2 than you will ever get out of it.
This is the same problem with the "hydrogen economy," as hydrogen is an energy carrier, not an energy source, and you still need something to generate the hydrogen in the first place (and do so in a way that is economically feasible). The only way to do this efficiently without fossil fuels is likely to be nuclear, but I digress.
A better approach to mitigate the carbon would be to engineer an organism that can use it. In fact, on Earth, most forms of the chlorophyll molecule are more efficient at higher CO2 concentrations. Also, the vast majority of oxygen-producing photosynthesis on the planet comes from phytoplankton in the ocean, and phytoplankton is one organism whose chlorophyll molecules perform better at higher CO2 concentrations.
Unfortunately though, phytoplankton growth is limited by the availability of iron, so some plans for dealing with CO2 on our own planet involve fertilizing the ocean using large amounts of iron to cause massive phytoplankton blooms. These blooms would then pull in the CO2 from the air and presumably die, falling to the bottom where their bodies would entomb the CO2 from the air. It's essentially what happens naturally, but this would be supercharging it.
So for your civilization to mitigate the CO2 in their atmosphere, rather than attempt to use the CO2 as an energy source, which would be by definition impossible, it might make more sense for them to engineer phytoplankton that can efficiently pull CO2 out of the atmosphere, preferably in a way that doesn't require some kind of fertilizer (such as iron). There is a danger here though, as if the phytoplankton grows unchecked, it could pull too much CO2 from the atmosphere in a relatively short amount of time (on the order of decades or less), and cause a crash of the ecosystem. Not only would pulling too much CO2 out of the air potentially cause cooling, but it could starve plants on a wide-scale and disrupt the carbon cycle, and worse, collapse the food web.