They can absorb CO₂, but what are they going to do with it?
Rubisco will happily grab CO₂ and bind it provided ribose 1-5 biphosphate is available, and provided the plant has energy, (from ATP, from sugar) it can make that.
But where will this plant get that energy. It has to come from somewhere. There are alternative sources of energy. Animals eat stuff, and get energy from their food (and so don't need to fix CO₂). There are some plants to do the same, they get energy by stealing it from other plants, and so don't need chlorophyll in their leaves.
There are chemoautotrophs, who get the energy to fix CO₂ from their environment. But they normally live in "extreme" environments, such as near undersea volcanic vents, and they are bacteria, not plants.
Perhaps your best option is something like Pterospora. This is a fungi parasite. It acquires energy from a fungus. It is possible to imagine a plant that fixes CO₂ in its leaves, but this is powered by energy stolen from a fungal partner. The fungus in turn gets energy by breaking down detritis that is trapped in the cave. One could even imagine a symbiotic relationship in which the plant provides something back to the fungus, perhaps an effective way of distributing spores. Or perhaps the fungus takes the polypeptide in the death material, decomposes it to provide ATP that would normally be produced by chlorophyll, and takes some of the sugars that the plant makes from CO₂