Multivariate count regression involves simultaneously associating multiple covariates with the observed counts across several response categories. For gut microbiome data from US immigrants, interest may lie in how acculturation is associated with gut bacterial abundances. The natural model for such data is the Dirichlet-multinomial (DM) distribution because of its ability to accommodate extra-multinomial variation in the observed counts. Unfortunately, the DM distribution falls outside the exponential family, complicating parameter estimation.  When performing variable selection with complex penalty functions, such as the sparse group lasso where coefficients for each covariate across the response categories are grouped together, optimization of the objective function is further challenged.  Here, we introduce Dominating Hyperplane Regularization (DHR) as a stable method of optimization. Based on the majorization-minimization framework, a suitable surrogate for the penalty function is used during optimization. In the case of gut microbiome data, where bacterial counts are associated through a taxonomic tree, we can use Dirichlet-Tree multinomial regression and introduce novel taxonomic tree-guided penalty functions that leverage known relationships among taxa. We study the performance of DHR for these novel models through simulation and on real world data from the Hispanic Community Health Study/Study of Latinos.