Babies born with low and very low birthweights -- i.e., birthweights below 2,500 and 1,500 grams, respectively -- have an increased risk of complications compared to other babies, and the proportion of babies with a low birthweight is a common metric used when evaluating public health in a population. While many factors increase the risk of a baby having a low birthweight, many can be linked to the mother's socioeconomic status, which in turn contributes to large racial disparities in the incidence of low weight births. Here, we employ Bayesian statistical models to analyze the proportion of babies with low birthweight in Pennsylvania counties by race/ethnicity. Due to the small number of births -- and low weight births -- in many Pennsylvania counties when stratified by race/ethnicity, our methods must walk a fine line. On one hand, leveraging spatial structure can help improve the precision of our estimates. On the other hand, we must be cautious to avoid letting the model overwhelm the information in the data and produce spurious conclusions. As such, we first develop a framework by which we can measure (and control) the informativeness of our spatial model. After demonstrating the properties of our framework via simulation, we analyze the low birthweight data from Pennsylvania and examine the extent to which the commonly used conditional autoregressive model can lead to oversmoothing. We then reanalyze the data using our proposed framework and highlight its ability to detect (or not detect) evidence of racial disparities in the incidence of low birthweight.