The rates of cell growth, division, and carbon loss of microbial populations are key parameters for understanding how organisms interact with their environment and how they contribute to the carbon cycle. However, the invasive nature of current analytical methods has hindered efforts to reliably quantify these parameters. In recent years, size-structured matrix population models (MPMs) have gained popularity for estimating division rates of microbial populations by mechanistically describing changes in microbial cell size distributions over time. Motivated by the mechanistic structure of these models, we employ a Bayesian approach to extend size-structured MPMs to capture additional biological processes describing the dynamics of a marine phytoplankton population over the day-night cycle. Our Bayesian framework is able to take prior scientific knowledge into account and generate biologically interpretable results. Using data from an exponentially growing laboratory culture of the cyanobacterium Prochlorococcus, we isolate respiratory and exudative carbon losses as critical parameters for the modeling of their population dynamics. The results suggest that this modeling framework can provide deeper insights into microbial population dynamics provided by size distribution time-series data.