Inertial confinement fusion (ICF) experiments create a unique laboratory environment in which thermonuclear fusion reactions occur within a plasma, with conditions comparable to stellar cores and the early universe. In contrast, accelerator-based measurements must compete with bound electron screening effects and beam stopping when measuring fusion cross sections at nucleosynthesis-relevant energies. Therefore, ICF experiments are a natural place to study nuclear reactions relevant to nuclear astrophysics. However, analysis of ICF-based measurements must address its own set of complicating factors. These include: the inherent range of reaction energies, spatial and temporal thermal temperature variation, and kinetic effects such as species separation. In this work we examine these phenomena and develop an analysis to quantify and, when possible, compensate for their effects on our inference. Error propagation in the analyses are studied using synthetic data combined with Markov Chain Monte Carlo (MCMC) machine learning. The novel inference techniques will aid in the extraction of valuable and accurate data from ICF-based nuclear astrophysics experiments.