Longitudinal microbiome studies often seek to separate background temporal variation from nonlinear perturbation-associated dynamics while accounting for the count-compositional nature of sequencing data. We introduce \textit{MultiAddGPs}, a semi-parametric Bayesian multinomial logistic-normal framework that combines linear effects with additive Gaussian process components. We first prove that MultiAddGPs remain within the Marginally Latent Matrix-T Process class (MLTP) and derive the collapsed and uncollapsed representations needed for scalable inference with the Collapse--Uncollapse (CU) sampler. We then develop a back-sampling procedure for efficient posterior inference on the individual additive components. We further develop approximate maximum marginal likelihood and penalized maximum marginal likelihood procedures for hyperparameter estimation in this broader class of models. In simulations, MultiAddGPs provides accurate posterior inference and calibrated uncertainty while scaling more effectively than existing GP-based additive alternatives. Applications to two longitudinal microbiome studies demonstrate how the model separates background temporal variation from intervention-associated dynamics, including travel-associated perturbations in a human gut time series and starvation effects in an artificial gut system. Our methods are implemented in the \textit{fido} package on CRAN.

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