Many scientific fields collect longitudinal count compositional data. Each observation is a multivariate count vector, where the total counts are arbitrary, and the information lies in the relative frequency of the counts. Multiple authors have proposed Bayesian Multinomial Logistic-Normal Dynamic Linear Models (MLN-DLMs) as a flexible approach to modeling these data. However, adoption of these methods has been limited by computational challenges. This article develops an efficient and accurate approach to posterior state estimation, called $\textit{Fenrir}$. Our approach relies on a novel algorithm for MAP estimation and an accurate approximation to a key posterior marginal of the model. As there are no equivalent methods against which we can compare, we also develop an optimized Stan implementation of MLN-DLMs. Our experiments suggest that Fenrir can be three orders of magnitude more efficient than Stan and can even be incorporated into larger sampling schemes for joint inference of model hyperparameters. Our methods are made available to the community as a user-friendly software library written in C++ with an R interface.