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The steadily increasing amount of atmospheric carbon dioxide (CO$_2$) is affecting the global climate system and threatening the long-term sustainability of Earth's ecosystem. In order to better understand the sources and sinks of CO$_2$, NASA operates the Orbiting Carbon Observatory-2 & 3 satellites to monitor CO$_2$ from space. These satellites make passive radiance measurements of the sunlight reflected off the Earth's surface in different spectral bands, which are then inverted in an ill-posed inverse problem to obtain estimates of the atmospheric CO$_2$ concentration. In this work, we propose a new CO$_2$ retrieval method that uses known physical constraints on the state variables and direct inversion of the target functional of interest to construct well-calibrated frequentist confidence intervals based on convex programming. We compare the method with the current operational retrieval procedure, which uses prior knowledge in the form of probability distributions to regularize the problem. We demonstrate that the proposed intervals consistently achieve the desired frequentist coverage, while the operational uncertainties are poorly calibrated in a frequentist sense both at individual locations and over a spatial region in a realistic simulation experiment. We also study the influence of specific nuisance state variables on the length of the proposed intervals and identify certain key variables that can greatly reduce the final uncertainty given additional deterministic or probabilistic constraints, and develop a principled framework to incorporate such information into our method.